1 Introduction

Baire [1] proved three properties of a separately continuous function \({f:{{\mathbb {R}}}^2\rightarrow {{\mathbb {R}}}}\) which are known in present day as three Baire theorems.

Theorem 1.1

(the Baire horizontal theorem [2]). Let X be a topological space, Y be a first countable space at some point \(b\in Y\), Z be a metrizable space and \(f:X\times Y\rightarrow Z\) be a separately continuous function. Then there exists a residual set A in X such that f is continuous at (a, b) for any \(a\in A\).

Theorem 1.2

(the Baire curve theorem [3]). Let X be a topological space, Y and Z be metrizable spaces, \(f:X\times Y\rightarrow Z\) be a separately continuous function and \(g:X\rightarrow Y\) be a continuous function. Then there exists a residual set A in X such that f is continuous at \(\big (a,g(a)\big )\) for any \(a\in A\).

Theorem 1.3

(the Baire projection theorem [2]). Let X be a topological space, Y be a second countable space, Z be a metrizable space and \(f:X\times Y\rightarrow Z\) be a separately continuous function. Then there exists a residual set A in X such that f is continuous at \(\big (a,b\big )\) for any \(a\in A\) and \(b\in B\). In particular, the projection \({\text {pr}}_XD(f)\) on X of the discontinuity point set D(f) of f is a meager subset of X.

Of course, we give more general (but not the most general) formulations of the Baire theorems which were obtained by other authors. In fact, many mathematicians generalized Baire’s results during XX century. But the approach by M. K. Fort plays an essential role in our paper.

In 1955  Fort [4] introduced the notion of a categorically related topologies: a topology \({{\mathcal {T}}}'\) on a set Y is categorically related to another topology \({{\mathcal {T}}}\) on Y if any \({{\mathcal {T}}}\)-continuous map \(f:X\rightarrow Y\) defined on a topological space X is \({{\mathcal {T}}}'\)-continuous at each point of a residual subset of X. He gave some necessary condition (it is called condition \((\alpha )\) in [4]) of topology to be a categorically related to another one. Fort’s results imply, in particular, that for a locally compact metrizable space Y and a separable metrizable space Z the compact-open topology on C(Y, Z) is categorically related to the pointwise topology (=the point-open topology =the topology of the pointwise convergence), and then every separately continuous function \(f:X\times Y\rightarrow Z\) is continuous on \(A\times Y\) for some residual set A in X. Another application of Fort’s theorem is related to upper and lower continuity of multifunctions: for any topological space X and separable (and locally compact) metrizable space Y every lower (upper) semi-continuous closed-valued multifunction \(F:X\multimap Y\) i upper (lower) continuous at each point of a residual subset of X (in the earlier papers [5] Fort proved a similar result for compact-valued multifunctions). In this paper we will say that a topology \({{\mathcal {T}}}'\) on a set Y is categorically related to a topology \({{\mathcal {T}}}\) on Y with respect to a space X if any \({{\mathcal {T}}}\)-continuous map \(f:X\rightarrow Y\) is \({{\mathcal {T}}}'\)-continuous at each point of a residual subset of X.

The first Fort’s corollary on separately continuous functions generalizes the result of Baire [1] (for \(f:{{\mathbb {R}}}^2\rightarrow {{\mathbb {R}}}\)) and Hahn [6] (for complete metric space X and metric compact Y). It states that the projection onto X of the discontinuity point set D(f) of a function f is meager in X. Now it is known as the Baire projection theorem. Later Calbix and Troallic [2] showed that the assumptions of the locally compactness of Y and the separableness of Z can be omitted and proved Theorems 1.1 and 1.3. The most powerful version of the Baire projection theorem (for a countable Čech complete space X and a \(\sigma \)-compact and locally compact space Y) was proved by Namioka [7]. To date, an increasing number of new versions of the Baire projection theorem continue to emerge: [8,9,10,11,12,13,14,15,16,17,18,19]. In particular, Namioka and co-Namioka spaces was investigated in these articles. In our terminology this notions can be defined in the following way. A space X is a Namioka space if for any compact Y the uniform topology (=the topology of the uniform convergent =the topology which is generated by the supremum-norm =the compact-open topology) on C(Y) is categorically related to the pointwise topology on C(Y) with respect to X. A compact space Y is a co-Namioka space if the uniform topology on C(Y) is categorically related to the pointwise topology on C(Y).

In the papers by Kenderov [20, 21] and Debs [11] one can find some generalization of the second Fort’s corollary on semi-continuous multifunctions. The most interesting for us is article [20] where the author proved, that for a Čech complete space X, a metric space Y any metrically upper (lower) semi-continuous multifunction \(F:X\multimap Y\) is metrically lower (upper) semi-continuous at each pint of a residual subset of X. Here the metrical lower and upper semi-continuity means the continuity with respect to the lower and upper Hausdorff hemimetrics on Y: \(d_{H}^-(A,B)=\sup _{x\in A}\inf _{y\in B}d(x,y)\) and \(d_H^+(A,B)=d_H^-(B,A)\), \(A,B\in 2^Y{\setminus }\{\varnothing \}.\) In fact, Kenderov [20] proved a more general result about an arbitrary hemimetric spaces (=pseudometric spaces without the symmetry axiom). The proof of the following result is easily obtained from [20, Basic lemma] applying to the function \(d\big (f(x),f(y)\big )\), \(x,y\in X\).

Theorem 1.4

(Kenderov, [20]). Let X be a Čech complete space, (Y, d) be a hemimetric space and \(f:X\rightarrow Y\) be a d-continuous function. Then there is a residual A in X such that f is \(d'\)-continuous at every point of A with respect to the hemimetric \(d'(x,y)=d(y,x)\), \(x,y\in Y\). Therefore, the topology \({{\mathcal {T}}}_{d'}\) on a set Y is categorically related to a topology \({{\mathcal {T}}}_d\) on Y with respect to X.

The appearance of the fundamental works by Fréchet [22, 23] and Hausdorff [24] at the beginning of XX century led to the implementation of the notions of metric spaces and topological spaces into the worldwide mathematical life. Almost immediately in the articles by Niemytzki [25, 26] and Wilson [27, 28] two generalizations of metric spaces appear: a quasi-metric (=metric without the symmetry axiom) and a semi-metric (=metric without the triangle inequality). In particular, Niemytzki [26] proved that any semi-metrizable or quasi-metrizable compact is metrizable. Wilson [27] proved that for any second countable \(T_1\)-space X there exists a quasi-metric d such that \(x_n\rightarrow a\) in X if and only if \(d(x,x_n)\rightarrow 0\) for any sequences \((x_n)_{n=1}^\infty \) in X, that is every second countable \(T_1\)-space is quasi-metrizable. Further, according to the modern tradition we write the prefixes quasi-, pseudo-, semi-, hemi-, pre- jointly with the main word.

Many other generalizations of metrics have been appearing during XX century: pseudometrics, ultrametrics, premetrics, hemimetrics, metametrics e.t.c. (see, for example [29,30,31,32,33,34,35,36,37,38]). Sometimes the terminology was duplicated: quasimetric=\(\Delta \)-metric, semimetric=symetric, o-metric=premetric, o-metriz able=premetrizable\(\ne \)metrizable in the wide sense. And sometimes collisions with the terminology appeared. The most unhappy notion became the good Wilson’s therm “quasimetric”. At the moment there are at least three meanings of this term: quasimetric in Wilson’s sense [28] =metric without the axiom of symmetry =\(T_1\)-hemimetric in our terminology, quasimetric in Goubault-Larrecq’s sense [37] =\(T_0\)-hemimetric in our terminology and in the Fixed Point Theory (see, for example, [35, 36]). We are facing with the completely different meaning: quasimetric in FPT’s sense =metric with the relaxed triangle inequality (\(d(x,y)\le r(d(x,z)+d(z,y))\) for some \(r\ge 1\)). Therefore, in this paper we will try to avoid the term “quasimetric” (and if we do that then we understand the quasimetric in Wilson’s sense)

Through the present paper we use the notions of hemimetric and hemimetrizability (see [37]). A function \(d:X\times X\rightarrow [0;+\infty )\) is called a hemimetric if \(d(x,x)=0\) and \(d(x,y)\le d(x,z)+d(z,y)\) for any \(x,y,z\in X\). A topological space X is said to be hemimetrizable if its topology coincides with the standard open ball topology for some hemimetric. The essential part of the paper can be divided into three directions:

  1. (I)

    General hemimetric spaces (sec. 2–5);

  2. (II)

    Separately continuous functions ranged in hemimetrizable spaces (sec. 6–7);

  3. (III)

    Generalization of the Kenderov theorem (sec. 8–10).

In (I) we give some definitions, facts and examples concerning with the general hemimetric spaces. But we also obtain some new results. In particular, we show the existence of the universal hemimetric space \(\ell _\infty ^{^{_\oplus }}(T)\) (Theorem 4.3) which allow to prove easily the hemimetrizability of a premetric space with the weak triangle inequality (Theorem 5.1).

The study of separately continuous function ranged in non-metrizable spaces was started respectively recently (see [39] and its references). Therefore, there are many open questions in this area. In (II) we are inspired mainly by article [39], in which the authors proved that the Baire horizontal theorem holds for a separately continuous function ranged in the Sorgenfrey line. We generalize this result for a function ranged in a regular hemimetrizable space (Theorem 7.3) and obtain some version of the Baire curves theorem in this situation (Theorem 7.2).

In (III), generalizing the Kenderov theorem, we go to the same direction as Christensen, Saint-Raymond, Talagrand, Bousiad and other mathematicians who generalized the Namioka theorem. In particular, we generalize Theorem 1.4 to the case where X is a \(\beta \)-unfavorable spaces for the Christensen \(\sigma \)-game (Theorem 9.1). Remark that the Sorgenfrey line is \(\alpha \)-favorable for the Saint-Raymond \(\sigma \)-game which is very close to the Christensen game. But the Sorgenfrey line is \(\beta \)-favorable for the Christensen \(\sigma \)-game (see Example 8.1). Moreover, Example 9.2 shows that the Kenderov theorem cannot be generalized to the Sorgenfrey line which is \(\alpha \)-\(\sigma \)-favorable space for the Saint-Raymond game.

2 Generalized metrics

Definition 1

Let X be a set and \(d:X^2\rightarrow [0,+\infty ]\) be a function. We use the following terminology, notations and properties.

  • \(B(x,\varepsilon )=B_d(x,\varepsilon )=\bigl \{y\in X:d(x,y)<\varepsilon \bigr \};\) (open ball)

  • \(B[x,\varepsilon ]=B_d[x,\varepsilon ]=\bigl \{y\in X:d(x,y)\le \varepsilon \bigr \};\) (closed ball)

  • U is called a d-neighborhood of x if there exists \(\varepsilon >0\) such that \(B(x,\varepsilon )\subseteq U\).

  • G is called d-open provided G is a d-neighborhood of each point \(x\in G\).

  • F is called d-closed if \(X\setminus F\) is d-open.

  • \({{\mathcal {T}}}_d=\{G: G\) is a d-open subset of \(X\}\); (open ball topology)

  • \(d'\) is defined by \(d'(x,y)=d(y,x)\) for any \(x,y\in X\); (dual function)

  • \(\widehat{d}\) is defined by \(\widehat{d}(x,y)=\max \{d(x,y),d(y,x)\}\) for any \(x,y\in X\);

  • (T) \(d(x,x)=0\) for any \(x\in X\);

  • \((T_0)\) \(\bigl (d(x,y)=0\wedge d(y,x)=0\bigr )\Rightarrow (x=y)\) for any \(x,y\in X\);

  • \((T_1)\) \(\bigl (d(x,y)=0\bigr )\Rightarrow (x=y)\) for any \(x,y\in X\);

  • \((\Delta )\)\(d(x,y)\le d(x,z)+d(z,y)\) for any \(x,y,z\in X\) (triangle inequality)

  • (S) \(d(x,y)=d(x,y)\) for any \(x,y\in X\); (symmetry)

  • (CBP) \(B_d[x,\varepsilon ]\) is d-closed for any \(x\in X\) and \(\varepsilon >0\); (closed ball property)

The proof of the following proposition is straightforward (see [37]).

Proposition 2.1

Let X be a set and \(d:X\times X\rightarrow [0,+\infty ]\) be a function. Then

(i):

\((T)\Leftrightarrow {{\mathcal {T}}}_d\) is a topology on X;

Furthermore, if d has property (T), then

(ii):

\((T_0)\Leftrightarrow (X,{{\mathcal {T}}}_d)\) is a \(T_0\)-space;

(iii):

\((T_1)\Leftrightarrow (X,{{\mathcal {T}}}_d)\) is a \(T_1\)-space;

(iv):

\((\Delta )\Rightarrow \) \(B_d(a,\varepsilon )\) is d-open for any \(a\in X\) and \(\varepsilon >0\);

(v):

\((\Delta )\Rightarrow \) \(B_d[a,\varepsilon ]\) is \(d'\)-closed for any \(a\in X\) and \(\varepsilon >0\);

(vi):

d has \((T_0)\Leftrightarrow \) \(\widehat{d}\) has \((T_1)\).

If, moreover, d has properties (T) and \((\Delta )\), then

(vii):

d has \((CBP)\Leftrightarrow \) for any \(x\in X\) the function \(d^x\) is \({{\mathcal {T}}}_d\)-continuous.

where \(d^x:X\rightarrow \overline{{{\mathbb {R}}}}\) is defined by \(d^x(y)=d(x,y)\) for any \(x,y\in X\).

Definition 2

A function \(d:X\times X\rightarrow [0,+\infty )\) is called

  • a premetric on X if property (T) holds.

  • a hemimetric on X if properties (T),\((\Delta )\) hold;

  • a \(T_0\)-hemimetric on X if properties (T), \((\Delta )\), \((T_0)\) hold;

  • a \(T_1\)-hemimetric on X if (T),\((\Delta )\), \((T_1)\) hold;

  • a pseudometric on X if properties (T), \((\Delta )\), (S) hold;

  • a metric on X if properties (T), \((\Delta )\), (S), \((T_1)\) hold;

  • a hemimetric with CBP on X if properties (T), \((\Delta )\), (CBP) hold.

We add the world “extended” to the previous terms for a function \({d:X{\times } X{\rightarrow }[0,+\infty ]}\) with the same properties. A pair (X, d) is called a metric (premetric, hemimetric, and so on) space if d is a metric (premetric, hemimetric and so on) on X.

Proposition 2.2

Let (X, d) be a premetric space. Then \({\widehat{d}}\) has (S). In particular,

(i):

if d is a hemimetric then \({\widehat{d}}\) is a pseudometric;

(ii):

if d is a \(T_0\)-hemimetric then \({\widehat{d}}\) is a metric;

Definition 3

Let \((X,{{\mathcal {T}}})\) be a topological space and d be an extended premetric. We say that d generates \({{\mathcal {T}}}\) if \({{\mathcal {T}}}\) consists of all d-open sets (that is \({{\mathcal {T}}}={{\mathcal {T}}}_d\)). A topological space X is called metrizable (premetrizable, hemimetrizable and so on) if there exists a metric (premetric, hemimetric and so on) d on X which generates the topology of X.

Using Proposition 2.1 we obtain the following.

Proposition 2.3

Let X be a topological space.

(i):

X is a hemimetrizable \(T_0\)-space if and only if it is \(T_0\)-hemimetrizable.

(ii):

X is a hemimetrizable \(T_1\)-space if and only if it is \(T_1\)-hemimetrizable.

(iii):

X is a pseudometrizable \(T_0\)-space if and only if it is metrizable.

(iv):

If X is a hemimetrizable space with CBP, then it is \(T_{3\frac{1}{2}}\)-space.

Here \(T_{3\frac{1}{2}}\) means the complete regularity without \(T_1\). The proof of the following proposition is straightforward.

Proposition 2.4

Let \(d:X\times X\rightarrow [0;+\infty ]\) be a function which have one of the properties (T), \((T_0)\), \((T_1)\), \((\Delta )\), (S), (CBP) and

$$\begin{aligned} d_1(x,y)=\left\{ \begin{array}{ccl} \frac{d(x,y)}{1+d(x,y)}&{},&{}\text { if }d(x,y)<\infty ;\\ 1&{},&{}\text { if }d(x,y)=\infty ,\\ \end{array} \right. \qquad \text { for any }x,y\in X. \end{aligned}$$

Then \(d_1:X\times X\rightarrow [0;1]\) has the same property. Furthermore, if d is an extended premetric, then d and \(d_1\) generate the same topologies.

The previous proposition shows that for the study of topological notions generated by extended hemimetrics it is sufficient to consider hemimetrics only. But extended hemimetrics are very suitable for some examples of hemimetrizable spaces.

3 Examples of hemimetrizable and non-metrizable spaces

We start this section with the exact formulation of the Niemytzki theorem [26, Statz III, S.670].

Theorem 3.1

(Niemytzki). Every countably compact Hausdorff hemimetrizable space is metrizable.

Example 3.2

The topologies of the Alexandroff double circle [40, 3.1.26] and the two arrows space [40, 3.10.C] are generated by some natural premetrics. But these spaces are not hemimetrizable by Theorem 3.1.

The following theorem is a “hemi” analogue of the Urysohn characterization of a separable metrizable space (see [37, Theorem 6.3.13], [27, section 7, Theorem III, p.681] or [33, Corollary 3, p.211])

Theorem 3.3

(Wilson). Every second countable space is hemimetrizable.

The following example shows that the Niemytzki theorem is false for a \(T_1\)-compact.

Example 3.4

Let \(X=({{\mathbb {N}}},{{\mathcal {T}}})\), where \({{\mathcal {T}}}=\big \{G\subseteq {{\mathbb {N}}}:|{{\mathbb {N}}}{\setminus } G|<\aleph _0\big \}\cup \{\varnothing \}\). Then X is a compact \(T_1\)-space but not Hausdorff (and so it is non-metrizable) [40, 1.2.6.]. On the other hand, \(|{{\mathcal {T}}}|=\aleph _0\) and then X is hemimetrizable by Theorem 3.3.

The following proposition is straightforward (see [33, Corollary 3, p.211])

Proposition 3.5

Let (X, d) be a premetric space such that the identity map \({{\text {id}}:X\rightarrow X}\) is \({{\mathcal {T}}}_{d}\)-\({{\mathcal {T}}}_{d'}\)-continuous. Then \({{\mathcal {T}}}_d={{\mathcal {T}}}_{{\widehat{d}}}\). In particular, if d is \(T_0\)-hemimetric (hemimetric) then \((X,{{\mathcal {T}}}_d)\) is metrizable (pseudometrizable)

The Sorgenfrey line \({{\mathbb {L}}}\) is an example of a normal hemimetrizable and non-metrizable space which is not second countable (it is well known that \(w({{\mathbb {L}}})=\mathfrak {c}\)).

Example 3.6

The Sorgenfrey line \({{\mathbb {L}}}\) (see [40, 1.2.2.]) is the set \({{\mathbb {R}}}\) equipped with the topology with the base \(\big \{[a;b):a<b\big \}\). It is hemimetrizable by the hemimetric

$$\begin{aligned} d(x,y)=\left\{ \begin{array}{ccl} y-x&{},&{}\text { if }x\le y,\\ 1&{},&{}\text { otherwise}, \end{array} \right. \text { for any }x,y\in {{\mathbb {R}}}. \end{aligned}$$

Example 3.7

The segment \(X{=}[0;1]\) with the topology \({{\mathcal {T}}}{=}\big \{(x;1]:x{\in } X\big \}{\cup }\{X\}\) is a compact \(T_0\)-space which is not a \(T_1\)-space. Therefore, it is not pseudometrizable by Proposition 2.3(ii). On the other hand, it is hemimetrizable by the hemimetric

$$\begin{aligned} d(x,y)=\left\{ \begin{array}{ccl} 0&{},&{}\text { if }x\le y,\\ 1&{},&{}\text { otherwise}, \end{array} \right. \text { for any }x,y\in X. \end{aligned}$$

Example 3.8

The connected doubleton \(\{0,1\}_0\) is the set \(\{0,1\}\) equipped by the topology \({{\mathcal {T}}}=\{\varnothing ,\{1\},\{0,1\}\}\) (so, 0 is the unique non-isolated point of \(\{0,1\}_0\)). This space is compact, \(T_0\), not \(T_1\) and hemimetrizable by the hemimetric

$$\begin{aligned} d(x,y)=\left\{ \begin{array}{ccl} 1&{},&{}\text { if }x=1\text { and } y=0,\\ 0&{},&{}\text { otherwise}, \end{array} \right. \text { for any }x,y\in \{0,1\}. \end{aligned}$$

Note, that the connected doubleton is a subspace of the space X from Example 3.7.

Example 3.9

The Bing plane (see [40, 6.1.6.]) is a non-metrizable second countable (and hence hemimetrizable by Theorem 3.3) Hausdorff countable connected space.

Example 3.10

Let \(X={{\mathbb {R}}}\), \(A=\big \{\frac{1}{n}:n\in {{\mathbb {N}}}\big \}\) and the topology \({{\mathcal {T}}}\) of X be generated by the base \({{\mathcal {B}}}=\big \{(a;b):a,b\in {{\mathbb {Q}}}\big \}\cup \big \{(a;b){\setminus } A:a,b\in {{\mathbb {Q}}}\big \}\). Since \(|{{\mathcal {B}}}|=\aleph _0\), X is hemimetrizable by Theorem 3.3. On the other hand, X is a Hausdorff space which is not regular [40, 1.5.6.]. In particular, X is a hemimetrizable space without CBP.

A relation \(R\subseteq X\times X\) is called a preordering on X if for every \(x,y,z\in X\) the following conditions are hold: \((x,x)\in R\); if \((x,y)\in R\) and \((y,z)\in R\), then \((x,z)\in R\). If X is a topological space and R is closed in \(X\times X\) then R is called a closed relation.

The following two propositions are straightforward.

Proposition 3.11

If d is an extended hemimetric then the relation \(R=\big \{(x,y):d(x,y)<+\infty \big \}\) is a preordering.

Proposition 3.12

Let (X, d) be a hemimetric space and R be a preordering on X. Then the function

$$\begin{aligned} d_R(x,y)=\left\{ \begin{array}{ccl} d(x,y)&{},&{}\text { if }(x,y)\in R;\\ +\infty &{},&{}\text { otherwise},\\ \end{array} \right. \qquad \text { where }x,y\in X, \end{aligned}$$

is an extended hemimetric on X. Moreover, if d has CBP and R is closed in \((X,{{\mathcal {T}}}_d)\) then \(d_R\) is an extended hemimetric with CBP. In particular, for any metric d and any closed preordering R the function \(d_R\) is a quasimetric with CBP

We denote the hemimetric space \((X,d_R)\) from the previous proposition by \(X^R\). Note, if \(\le \) is the usual ordering of \({{\mathbb {R}}}\) and \(d(x,y)=|x-y|\), then \({{\mathbb {R}}}^{^\le }=({{\mathbb {R}}},d_{\le })\) is homeomorphic to the Sorgenfrey line \({{\mathbb {L}}}\).

Example 3.13

Let X be a nontrivial normed lattice with a partial ordering \(\le \) and \(d(x,y)=\Vert x-y\Vert \) for \(x,y\in X\). Then \(X^{^\le }=(X,d_{\le })\) is a \(T_1\)-hemimetric space with CBP which is not metrizable.

Proof

Since X is nontrivial, there is \(a\in X\) such that \(a\ge 0\) and \(a\ne 0\). Set \(L=\big \{ta:t\in {{\mathbb {R}}} \}\). Let \(f:{{\mathbb {L}}}\rightarrow L\subseteq X^{^\le }\), \(f(t)=ta\) for any \(t\in {{\mathbb {L}}}\). It is easy to see, that f is a homeomorphism. Thus, L (and then \(X^{^\le }\)) is not metrizable. The rest follows from Proposition 3.12. \(\square \)

In particular, the spaces \(\ell _p^{^\le }\), \(c_0^{^\le }\), \(\ell _p^{^\le }(T)\), \(L_p^{^\le }(\mu )\), \(C^{^\le }[a;b]\) are non-metrizable and \(T_1\)-hemimetrizable with CBP.

4 Heminormed spaces and the universal hemimetric space

Definition 4

Let X be a vector space and \(p:X\rightarrow [0;+\infty )\). A function p is called a heminorm if \(p(x+y)\le p(x)+p(y)\) and \(p(\lambda x)=\lambda p(x)\) for any \(x,y\in X\) and \(\lambda \ge 0\) (in other words, it is a positive sublinear functional). If, moreover, \(p(x)=0\) and \(p(-x)=0\) implies \(x=0\) for any \(x\in X\) then p is called a \(T_0\)-heminorm. A pair (X, p) is called heminormed space if p is a heminorm on a vector space X.

Obviously, for any heminorm p the natural distance \(d(x,y)=p(x-y)\) is a hemimetric. In the case where p is a \(T_0\)-heminorm we conclude that d is a \(T_0\)-hemimetric. Furthermore, in this case the function \(\Vert x\Vert _\infty =\max \big \{p(x),p(-x)\big \}\), \(x\in X\), is a norm on X and we say that a heminorm p generates the norm \(\Vert \,\cdot \,\Vert _\infty \). Note that we can consider another norms assigning with the heminorm p. For example, if \(s\ge 1\) then the function \(\Vert x\Vert _s=\Big (\big (p(x)\big )^s+\big (p(-x)\big )^s\Big )^{1/s}\), \(x\in X\), is also a norm on X (which is equivalent to the norm \(\Vert \,\cdot \,\Vert _\infty \)).

The most famous example of a heminorm is the Minkowski functional. But the next example is also very important for us.

Example 4.1

Let X be a normed lattice with a norm \(\Vert \,\cdot \,\Vert \). Let us define the function \({\Vert \,\cdot \, \Vert ^{^{_\oplus }}:X\rightarrow [0;+\infty )}\) by \(\Vert x\Vert ^{^{_\oplus }}=\Vert x^+\Vert \) for any \(x\in X\), where \(x^+=\sup \{x,0\}\) is the positive part of x. Then \(X^{^{_\oplus }}=(X,\Vert \,\cdot \, \Vert ^{^{_\oplus }})\) is a \(T_0\)-heminormed space.

Observe that \(x\le y\Leftrightarrow \Vert y-x\Vert ^{^{_\oplus }}=0\) for any \(x,y\in X\). Thus, a heminormed space is a generalization of a normed lattice, in some sense. But unfortunately, the heminorm \(\Vert \,\cdot \, \Vert ^{^{_\oplus }}\) does not generate the original norm of X. Indeed, let \(\Vert \,\cdot \, \Vert ^{^{_\oplus }}\) generate the norm \(\Vert x\Vert _\infty =\max \big \{\Vert x\Vert ^{^{_\oplus }},\Vert -x\Vert ^{^{_\oplus }}\big \}\), \(x\in X\). Since \(|x|=x^+\vee x^-\le x^++x^-=x^++(-x)^+,\) we obtain that \(\Vert x\Vert _\infty \le \big \Vert |x|\big \Vert =\Vert x\Vert \le \Vert x^+\Vert +\Vert x^-\Vert =\Vert x\Vert ^{^{_\oplus }}+\Vert -x\Vert ^{^{_\oplus }}\le 2\Vert x\Vert _\infty .\) Therefore, the norm \(\Vert \,\cdot \,\Vert _\infty \) is equivalent to the original norm \(\Vert \,\cdot \,\Vert \) but not equals. For example, in the case where the lattice X is an M-space we have that \({\Vert x\Vert =\max \{\Vert x^+\Vert ,\Vert x^-\Vert \}=\Vert x\Vert _\infty }\), but for L-spaces we have the opposite situation: \(\Vert x\Vert =\Vert x^+\Vert +\Vert x^-\Vert =\Vert x\Vert ^{^{_\oplus }}+\Vert -x\Vert ^{^{_\oplus }}=\Vert x\Vert _1\).

Observe that the space \({{\mathbb {R}}}^{^{_\oplus }}\) is \(T_0\) but not \(T_1\). Therefore, \({{\mathbb {R}}}^{^{_\oplus }}\) is not pseudometrizable. So, similarly to Example 3.13 we obtain the following.

Example 4.2

Let X be a nontrivial normed lattice. Then \(X^{^{_\oplus }}\) is a \(T_0\)-hemimetric space which is not pseudometrizable.

Therefore, \(\ell _p^{^{_\oplus }}\), \(c_0^{^{_\oplus }}\), \(\ell _p^{^{_\oplus }}(T)\), \(L_p^{^{_\oplus }}(\mu )\), \(C^{^{_\oplus }}[a;b]\) and so on are \(T_0\)-hemimetrizable but not pseudometrizable spaces. But one of them plays a special role in the class of hemimetric spaces. This space is \(\ell _\infty ^{^{_\oplus }}(T)\).

Definition 5

Let T be a nonempty set and \(\ell _\infty (T)\) be the space of bounded function \(x:T\rightarrow {{\mathbb {R}}}\) with the sup-norm \(\Vert x\Vert _\infty =\sup \nolimits _{t\in T}|x(t)|\).Denote \({\ell _\infty ^{^{_\oplus }}(T)=\big (\ell _\infty (T)\big )^{^{_\oplus }}}\), that is \(\ell _\infty ^{^{_\oplus }}(T)=\big (\ell _\infty (T),\Vert \,\cdot \,\Vert _\infty ^{^{_\oplus }}\big )\), where \(\Vert x\Vert _\infty ^{^{_\oplus }}=\Vert x^+\Vert _\infty =\sup \nolimits _{t\in T}\max \big \{x(t),0\big \}\). Then \(\ell _\infty ^{^{_\oplus }}(T)\) is also a hemimetric space with the hemimetric

$$\begin{aligned} d_\infty ^{^{_\oplus }}(x,y)=\Vert x-y\Vert ^{^{_\oplus }}_\infty =\left\{ \begin{array}{ccl} 0&{},&{}\text { if }x\le y;\\ \sup \limits _{t\in T}\big (x(t)-y(t)\big )&{},&{}\text { otherwise},\\ \end{array} \right. \ \text {where} \ x,y\in \ell _\infty (T). \end{aligned}$$

This formula also defines the extended hemimetric on the space \({{\mathbb {R}}}^T\) of all real-valued functions. We denote the extended hemimetric space \(({{\mathbb {R}}}^T,d_\infty ^{^{_\oplus }})\) by \({{\mathbb {R}}}^T_{^\oplus }\)

Note, that in this case the heminorm \(\Vert \,\cdot \,\Vert _\infty ^{^{_\oplus }}\) generates the original norm \(\Vert \,\cdot \,\Vert _\infty \).

Theorem 4.3

Let (X, d) be a hemimetric space. Then there exists \(f:X\rightarrow \ell _\infty (X)\) such that \(d(x,y)=d_\infty ^{^{_\oplus }}\big (f(x),f(y)\big )\) for any \(x,y\in X\). Moreover, if X is \(T_0\), than f is an injection. Therefore, the hemimetric space \(\ell _\infty ^{^{_\oplus }}(T)\) is isometrically universal for the class of all \(T_0\)-hemimetric spaces of the cardinality \(\le |T|\).

Proof

Of course our idea is not so far from the wellknown proof of the metric universality of \(\ell _\infty (T)\). But we need to involve some asymmetry by removing of the modulus in the definition of the distance in \(\ell _\infty (T)\). So, let us fix an arbitrary point \(a\in X\) and define \({f:X\rightarrow {{\mathbb {R}}}^X}\) by

$$\begin{aligned} f(x)(t)=d(x,t)-d(a,t),\text { for any }x,t\in X. \end{aligned}$$

By the triangle inequality \((\Delta )\) we obtain that \(d(a,t)\le d(a,x)+d(x,t)\) and \({d(x,t)\le d(x,a)+d(a,t)}\). Then \(-d(a,x)\le f(x)(t)=d(x,t)-d(a,t)\le d(x,a) \) for any \(x,t\in X\). Therefore, \(f:X\rightarrow \ell _\infty (X)\).

Let \(x,y\in X\) and \(\alpha =f(x)-f(y)\). Then \(\alpha (t)= d(x,t)-d(y,t)\) for any \(t\in X\). But \(d(x,t)\le d(x,y)+d(y,t)\). Therefore, \(\alpha (t)\le d(x,y)\) for any \(t\in T\). On the other hand, \(\alpha (y)=d(x,y)\ge 0\). Thus, \(d_\infty ^{^{_\oplus }}(f(x),f(y))=\sup \nolimits _{t\in T}\alpha (t)=d(x,y)\).

Finally, suppose that X is \(T_0\) and show the injectivity of f. Fix distinct points \(x,y\in X\) and put \(\alpha =f(x)-f(y)\). Observe, that \(\alpha (y)=d(x,y)\) and \(\alpha (x)=-d(y,x)\). Consequently, by \((T_0)\) we conclude that \(d(x,y)>0\) or \(d(y,x)>0\), and so \(\alpha (t)\ne 0\) for some \(t\in \{x,y\}\). Therefore, \(\alpha \ne 0\) and then \(f(x)\ne f(y)\). \(\square \)

5 Hemimetrizability of a premetric space

In this sections we will use the universal hemimetric space \({{\mathbb {R}}}^T_{^\oplus }\) to the proof of the hemimetrizability of a premetric space with some weakening of the triangle inequality.

Definition 6

We say that \(d:X^2\rightarrow [0;+\infty ]\) has the weak triangle inequality if

$$\begin{aligned} ({\widetilde{\Delta }})\ \ \ \forall x_0\in X\ \forall \varepsilon>0\ \exists \delta >0\ \forall x,y\in X\ \Big |\ d(x_0,x)<\delta \Rightarrow d(x_0,y)<\varepsilon +d(x,y) \end{aligned}$$

Of course, the triangle inequality \((\Delta )\) implies \(({\widetilde{\Delta }})\) with \(\delta =\varepsilon \).

Theorem 5.1

Let d be a premetric on a set X which has the weak triangle inequality \(({\widetilde{\Delta }})\). Then \((X,{{\mathcal {T}}}_d)\) is a hemimetrizable space.

Proof

Let us define \(f:X\rightarrow {{\mathbb {R}}}^X_{^\oplus }\) by the formula

$$\begin{aligned} f(x)(t)=d(x,t),\ \ \ x,t\in X, \end{aligned}$$

and put

$$\begin{aligned} \varrho (x,y)=d^{^{_\oplus }}_\infty \big (f(x),f(y)\big ) \text { for any } x,y\in X. \end{aligned}$$

Obviously, the function \(\varrho :X\times X\rightarrow [0;+\infty ]\) has (T) and \((\Delta )\). By Proposition 2.4, it remains to show \({{\mathcal {T}}}_d={{\mathcal {T}}}_\varrho \).

First of all, we note that for any \(x,y\in X\)

$$\begin{aligned}&d(x,y)=f(x)(y)-f(y)(y)\le \sup \limits _{t\in X}\big (f(x)(t)-f(y)(t)\big )=d^{^{_\oplus }}_\infty \big (f(x), f(y)\big )=\varrho (x,y). \end{aligned}$$

Therefore, \(B_\varrho (x,\varepsilon )\subseteq B_d(x,\varepsilon )\), and then \({{\mathcal {T}}}_d\subseteq {{\mathcal {T}}}_\varrho \).

To prove the inverse inclusion \({{\mathcal {T}}}_\varrho \subseteq {{\mathcal {T}}}_d\) we consider \(G\in {{\mathcal {T}}}_\varrho \) and \(x_0\in G\). Then there is \(\varepsilon >0\) with \(B_\varrho (x_0,2\varepsilon )\subseteq G\). Choose \(\delta >0\) by \(({\widetilde{\Delta }})\) and let \(x\in B_d(x_0,\delta )\). Therefore, \(d(x_0,t)<\varepsilon +d(x,t)\). If \(f(x_0)\le f(x)\) then \(\varrho (x_0,y)=0<2\varepsilon \) else

$$\begin{aligned} \varrho (x_0,y)=\sup _{t\in T}\big (f(x_0)(t)-f(x)(t)\big )=\sup _{t\in T}\big (d(x_0,t)-d(x,t)\big )\le \varepsilon <2\varepsilon . \end{aligned}$$

Thus, \(B_d(x_0,\delta )\subseteq B_\varrho (x_0,2\varepsilon )\subseteq G\) and so \(G\in {{\mathcal {T}}}_d\). \(\square \)

Now we apply the previous theorem to show the hemimetrizability of the Niemytzki plane \({{\mathbb {P}}}\). Recall [40, 1.2.4], that \({{\mathbb {P}}}\) is the set \({{\mathbb {R}}}\times [0;+\infty )\) equipped with the topology with the base \({{\mathcal {B}}}=\Big \{U_\varepsilon (p):p\in {\mathbb {P}},\varepsilon >0\Big \}\), where

$$\begin{aligned} U_\varepsilon (p)= \left\{ \begin{array}{rcl} \Big \{(u,v)\in {{\mathbb {P}}}:(x-u)^2+(y-v)^2<\varepsilon ^2\Big \}&{}, &{}\text { if } y>0; \\ \\ \Big \{(u,v)\in {{\mathbb {P}}}:(x-u)^2+v^2<\varepsilon v\Big \}\cup \{p\}&{}, &{} \text { if }y=0. \end{array}\right. \end{aligned}$$

for any \(p=(x,y)\in {{\mathbb {P}}}\).

In the following example we construct a premetric with \(({{\widetilde{\Delta }}})\) which generates the topology of \({{\mathbb {P}}}\). So, taking into account Theorem 5.1, we can construct a natural hemimetric which generates the topology of the Niemytzki plane \({{\mathbb {P}}}\).

Example 5.2

The Niemytzki plain \({{\mathbb {P}}}\) is a hemimetrizable space.

Proof

During this proof \(\Vert \,\cdot \,\Vert \) means the Euclidian norm on \({{\mathbb {R}}}^2\). We define the function \(\varphi :{{\mathbb {P}}}\times {{\mathbb {P}}}\rightarrow [0;+\infty ]\) by

$$\begin{aligned} \varphi (p,q)= \left\{ \begin{array}{ccl} \Vert p-q\Vert &{}, &{}\text { if } y>0;\\ \frac{1}{v}(x-u)^2+v&{}, &{} \text { if }y=0\text { and }v>0;\\ 0&{}, &{} \text { if }y=v=0\text { and }x=u;\\ +\infty &{}, &{} \text { if }y=v=0\text { and }x\ne u;\\ \end{array}\right. \end{aligned}$$

for any \(p=(x,y)\) and \(q=(u,v)\) in \({{\mathbb {P}}}\). Obviously, \(\varphi \) is an extended premetric on \({{\mathbb {P}}}\) and \(B_\varphi (p,\varepsilon )=U_\varepsilon (p)\) for any \(p\in {{\mathbb {P}}}\) and \(\varepsilon >0\). Therefore, \(\varphi \) generates the topology of \({{\mathbb {P}}}\). Denote \({{\mathbb {P}}}_+={{\mathbb {R}}}\times (0;+\infty )\) and \({{\mathbb {P}}}_0={{\mathbb {R}}}\times \{0\}\). Observe, that \(\varphi (p_0,q)\le \varphi (p_0,p)+\varphi (p,q)\) for any \(p_0,p\in {{\mathbb {P}}}_+\) and \(q\in {{\mathbb {P}}}\). But we have problems with the triangle inequality in the case \(p_0\in {{\mathbb {P}}}_0\). Therefore, we consider another premetric

$$\begin{aligned} d(p,q)= \left\{ \begin{array}{ccl} \varphi (p,q)&{}, &{}\text { if } y=0\text { and }\varphi (p,q)<1;\\ 1&{}, &{} \text { if }y=0\text { and }\varphi (p,q)\ge 1;\\ \frac{2}{y}\varphi (p,q)&{}, &{} \text { if }y>0\text { and }\varphi (p,q)<\frac{y}{2};\\ 1&{}, &{} \text { if }y>0\text { and }\varphi (p,q)\ge \frac{y}{2};\\ \end{array}\right. \end{aligned}$$

for any \(p=(x,y)\) and \(q=(u,v)\) in \({{\mathbb {P}}}\). Obviously, \(d:{{\mathbb {P}}}\times {{\mathbb {P}}}\rightarrow [0;1]\) has (T). Note that for any \(\varepsilon \in (0;1)\) and \(p\in {{\mathbb {P}}}_0\) we have \(B_d(p,\varepsilon )=B_\varphi (p,\varepsilon )\). On the other hand, if \(p=(x,y)\in {{\mathbb {P}}}_+\) then we can choose \(n\in {{\mathbb {N}}}\) with \(\frac{1}{n}<\frac{y}{2}<n\). Let us prove the inclusions \(B_\varphi (p,\frac{\varepsilon }{n})\subseteq B_d(p,\varepsilon )\subseteq B_\varphi (p,n\varepsilon )\). Let \(\varphi (p,q)<\frac{\varepsilon }{n}\). Then \(\varphi (p,q)<\frac{1}{n}<\frac{2}{y}\). Therefore, \(d(p,q)= \frac{2}{y}\varphi (p,q)\le n\cdot \frac{\varepsilon }{n}=\varepsilon \). On the other hand, let \(d(p,q)<\varepsilon <1\). Then \(d(p,q)=\frac{2}{y}\varphi (p,1)<\varepsilon \). So, \(\varphi (p,q)<\frac{y}{2}\cdot \varepsilon <n\varepsilon \). Thus, \({{\mathcal {T}}}_d={{\mathcal {T}}}_\varphi \).

By Theorem 5.1, it is enough to prove that the premetric d has the weak triangle inequality \(({\widetilde{\Delta }})\). Fix \(p_0=(x_0,y_0)\in {{\mathbb {P}}}\) and \(\varepsilon >0\). Put \(\delta =\min \big \{1,\frac{\varepsilon }{2}\big \}\). Consider points \(p=(x,y)\) and \(q=(u,v)\) in \({{\mathbb {P}}}\) with \(d(p_0, p)<\delta \) and prove that \(d(p_0,q)<\varepsilon +d(p,q).\) Of course, we may assume that the points \(p_0,p,q\) are distinct. On the other hand, if \(d(p,q)\ge 1\) then \(d(p_0,q)\le 1<\varepsilon +d(p,q)\). Thus, we assume that \(d(p,q)<1\). Moreover, since \(\delta \le 1\), we conclude \(d(p_0,p)<1\).

Let us consider the case \(y_0>0\). Therefore, \(d(p_0,p)=\frac{2}{y_0}\varphi (p_0,p)<\delta \). So, \(|y_0-y|\le \Vert p_0-p\Vert =\varphi (p_0,p)<\frac{\delta y_0}{2}\). In particular, \(|y_0-y|<\frac{y_0}{2}\) and so \(y>\frac{y_0}{2}>0\). Consequently, \(d(p,q)=\frac{2}{y}\varphi (p,q)\). Since \(p_0,p\in {{\mathbb {P}}}_+\), we have

$$\begin{aligned} \textstyle d(p_0,q)\le \frac{2}{y_0}\varphi (p_0,q) \le \frac{2}{y_0}\Big (\varphi (p_0,p)+\varphi (p,q)\Big ) =\frac{2}{y_0}\varphi (p_0,p)+\frac{y}{y_0}\cdot \frac{2}{y}\varphi (p,q)= \\ \textstyle =d(p_0,p)+\frac{y}{y_0}d(p,q)=d(p_0,p)+\frac{y-y_0}{y_0}d(p,q)+d(p,q). \end{aligned}$$

But \(\frac{y-y_0}{y_0}\le \frac{\delta }{2}\) and \(d(p,q)<1\). Thus, \(d(p_0,q)<\frac{3\delta }{2}+d(p,q)<\varepsilon +d(p,q)\).

Now we pass to the case \(y_0=0\). Let \(r=d(p_0,p)\). Since \(r<\delta <1\), we conclude \(r=\varphi (p_0,p)\). So, \(p\in U_\delta (p_0){\setminus }\{p_0\}\subseteq {{\mathbb {P}}}_+\) and then \(y>0\). Therefore, \(1>d(p,q)=\frac{2}{y}\varphi (p,q)\). Hence, \(\varphi (p,q)=\Vert p-q\Vert <\frac{y}{2}\). But \(|y-v|\le \Vert p-q\Vert <\frac{y}{2}\). Thus, \(v>\frac{y}{2}>0\). Observe, \(r=\varphi (p_0,p)=\frac{1}{y}(x_0-x)^2+y\). Then \(y\le r\) and \((x_0-y)^2+y^2=yr.\) Let \(p_1=(x_0,r)\) be the center of the circle \(U_{2r}(p_0)\). We consider the point \(p_2\) of the boundary of this circle such that \(p\in [p_1,p_2]\). Denote \({\text {dist}}(p,E)=\inf \nolimits _{t\in E}\Vert p-t\Vert \) for any nonempty set E in \({{\mathbb {R}}}^2\). By the geometrical thinking we have

$$\begin{aligned} {\text {dist}}(p,{{\mathbb {P}}}\setminus U_{2r}(p_0))=\Vert p-p_2\Vert =r-\alpha , \end{aligned}$$

where

$$\begin{aligned} \alpha =\Vert p-p_1\Vert =\sqrt{(x-x_0)^2+(y-r)^2}=\sqrt{\big ((x-x_0)^2+y^2\big )-2yr+r^2}= \\ \textstyle =\sqrt{yr-2yr+r^2}=\sqrt{r^2-yr}=\sqrt{\big (r-\frac{y}{2}\big )^2-\big (\frac{y}{2}\big )^2}\le r-\frac{y}{2}. \end{aligned}$$

Therefore, \({\text {dist}}(p,{{\mathbb {P}}}{\setminus } U_{2r}(p_0))=r-\alpha \ge \frac{y}{2}>\Vert p-q\Vert \). Hence, \(q\not \in {{\mathbb {P}}}\setminus U_{2r}(p_0)\) and then \(q\in U_{2r}(p_0)\). So, \( d(p_0,q)=\varphi (p_0,q)<2r<2\delta \le \varepsilon \le \varepsilon +d(p,q). \)

Thus, we have proved the weak triangle inequality for d and then by Theorem 5.1 there exists a hemimetric \(d_1\) which generates the topology of \({{\mathbb {P}}}\). \(\square \)

6 The preoscillation of a premetric-valued function

Definition 7

Let X be a topological space, (Y, d) be a premetric space, \({f:X{\rightarrow } Y}\) be a function, E be a non-empty subset of X and \(x\in X\). The preoscillation of f at x with respect to d is defined by

$$\begin{aligned} {{\widetilde{\omega }}}_{d,f}(x)=\inf _{U\in {{\mathcal {U}}}_x}{{\widetilde{\omega }}}_{d,f}(x,U),\text { where }{{\widetilde{\omega }}}_{d,f}(x,U)=\sup _{u\in U}d\bigl (f(x),f(u)\bigr ) \end{aligned}$$

and \({{\mathcal {U}}}_x\) is the system of all neighborhoods of x in X.

Proposition 6.1

Let X, Y be topological spaces, d be a premetric which generates the topology of X, \({f: X\rightarrow Y}\) be a function and \(x\in X\). Therefore,

(i):

if \({{\widetilde{\omega }}}_{d,f}(x)=0\) then f is continuous at x;

(ii):

if d is a hemimetric and f is continuous at x then \({{\widetilde{\omega }}}_{d,f}(x)=0\).

Proof

(i). Suppose that \({{\widetilde{\omega }}}_{d,f}(x)=0\). Let V be a d-open subset with \(f(x)\in V\). Then V ia a d-neighborhood of f(x). So, there is \(\varepsilon >0\) such that \(B_d\big (f(x),\varepsilon \big )\subseteq V\). Since \({{\widetilde{\omega }}}_{d,f}(x)=\inf \nolimits _{U\in {{\mathcal {U}}}_x}\omega _{d,f}(x,U)=0<\varepsilon \), there is \(U\in {{\mathcal {U}}}_x\) with \(\omega _{d,f}(x,U)<\varepsilon \) Therefore, \(f(U)\subseteq B_d\big (f(x),\varepsilon \big )\subseteq V\). Thus, f is continuous at x.

(ii). Let d be a hemimetric and f be a continuous function at x. Consider \(\varepsilon >0\). Proposition 2.1(iv) implies that \(V=B_d\big (f(x),\varepsilon \big )\) is a d-open neighborhood of f(x). Therefore, there exists \(U\in {{\mathcal {U}}}_x\) such that \(f(U)\subseteq V\). Consequently, \(0\le {{\widetilde{\omega }}}_{d,f}(x)\le {{\widetilde{\omega }}}_{d,f}(x,U)\le \varepsilon \) and so \(0\le {{\widetilde{\omega }}}_{d,f}(x)\le \varepsilon \) for any \(\varepsilon >0\). Thus, \({{\widetilde{\omega }}}_{d,f}(x)=0\). \(\square \)

It is well known that if Y is a metric space then Proposition 6.1 holds for the usual oscillation \(\omega _f(x)=\inf \nolimits _{U\in {{\mathcal {U}}}_x}\omega _f(U)\), where \(\omega _f(U)=\sup \nolimits _{x,x'\in U}d\big (f(x),f(x')\big ).\) Consequently, since \(\omega _f\) is upper semicontinuous, we have that the continuity point set of f is a \(G_\delta \)-set. The following example shows that this statement is not valid for functions ranged in the Sorgenfrey line. It follows that, in general, the pointwise oscillation \({\widetilde{\omega }}_f\) is not upper semicontinuous and Proposition 6.1 is false for the usual oscillation \(\omega _f\).

Example 6.2

Let C be the Cantor set in [0; 1], \((a_n,b_n)\) be nonempty intervals with \([0;1]{\setminus } C=\bigsqcup \nolimits _{n=1}^\infty (a_n;b_n)\), \({{\mathbb {L}}}\) be the Sorgenfrey line and \({h:C\rightarrow {{\mathbb {L}}}}\) such that \({h(x)=-x}\) for any \(x\in X\). Then the continuity point set C(h) of h is equal to the set \({C_0=\{1,a_1,a_2,\dots \}}\) which is not a \(G_\delta \)-set in C.

Proof

First of all, we note that \(C_0\) is a meager dense subset of the compact space C and then A is not \(G_\delta \)-set in C.

Let us prove that f is continuous at each point of \(C_0\). Fix \(a\in C_0\). Let V be a neighborhood of \(h(a)=-a\) in the Sorgenfrey line \({{\mathbb {L}}}\). Then there exists \(\varepsilon >0\) with \([-a;-a+\varepsilon \big )\subseteq V\). Put \(b=2\) if \(a=1\) and \(b=b_n\) if \(a=a_n\) for some \(n\in {{\mathbb {N}}}\). Set \(U=(a-\varepsilon ;b)\). Consider a point \(x\in C\cap U\). Since \(C\cap (a;b)=\varnothing \), we have \(x\not \in (a;b)\). Then \(x\in U{\setminus } (a;b)=(a-\varepsilon ;a]\). Thus, \(f(x)=-x\in [-a;-a+\varepsilon )\subseteq V\). Therefore, f is continuous at a.

Now we show that f is discontinuous at each point of \(C{\setminus } C_0\). Fix \(x_0\in C{\setminus } C_0\). Let \(V_0=[-x_0,+\infty )\). Then V is a neighborhood of \(h(x_0)=-x_0\) in \({\mathbb {L}}\). Consider a neighborhood U of \(x_0\) in C. Then there exists \(\varepsilon >0\) with \((x_0-\varepsilon ;x_0+\varepsilon )\cap C\subseteq U\). Since \(x_0\in C\setminus C_0\), it is easy to see that there exists \(x_1\in U\cap (x_0;x_0+\varepsilon )\cap C\). Consequently, \(h(x_1)=-x_1<-x_0\) and then \(h(x_1)\not \in V_0\). Therefore, \(h(U)\not \subseteq V_0\) for any neighborhood U of \(x_0\) in C. So, f is discontinuous at \(x_0\). \(\square \)

A function \(f:X\rightarrow Y\) is called quasicontinuous if for any \(x\in X\), any open sets U and V with \(x\in U\) and \(f(x)\in V\) there exists a non-empty open set \(U_1\subseteq U\) such that \(f(U_1)\subseteq V\). It is well known that the discontinuity point set of a quasicontinuous function ranged in a metrizable space is meager. But this is false for functions ranged in hemimetrizable spaces. Indeed, the function h from Proposition 6.2 is quasicontinuous but the discontinuity point set of h equals to the set \(C\setminus C_0\) which is not meager. We can construct a simpler example of such a function. Let \(f:{{\mathbb {L}}}\rightarrow {{\mathbb {L}}}\) be a function such that \(f(x)=-x\) for any \(x\in X\). Then f is quasicontinuous and the discontinuity point set equals to \({{\mathbb {L}}}\) which is a Baire space.

7 The Baire horizontal theorem and the Baire curves theorem for hemimetrizable spaces

Definition 8

Let X, Y and Z be topological spaces. For a function \(f:X\times Y\rightarrow Z\) we define \(f^x:Y\rightarrow Z\) and \(f_y:X\rightarrow Z\) by \({f^x(y)=f_y(x)=f(x,y)}\) for any \(x\in X\) and \(y\in Y\). A function f is called separately continuous if the functions \(f^x\) and \(f_y\) are continuous for any \(x\in X\) and \(y\in Y\).

Note, that \(T_3\) means the regularity without \(T_1\). More precisely, X is a \(T_3\)-space if for any closed set A in X and any \(b\in X\setminus A\) there are open sets U and V such that \(A\subseteq U\), \(b\in B\) and \(U\cap V=\varnothing \).

Lemma 7.1

Let d be a premetric which generates the topology of a \(T_3\)-space X and let \({{\bar{d}}:X^2\rightarrow [0;+\infty )}\) is defined by \({\bar{d}}(x,y)=\inf \Big \{r\ge 0:y\in \overline{B_d[x,r]}\Big \}. \) Then \({\bar{d}}\) is a premetric which generates the topology of X.

Proof

Firstly, observe that if \(y\notin \overline{B_d[x,r]}\) then \({\bar{d}}(x,y)\ge r\). Therefore, \(B_{{\bar{d}}}(x,r) \subseteq \overline{B_d[x,r]}\). Secondly, since \(y\in \overline{B_d[x,r]}\) with \(r=d(x,y)\), we conclude \({\bar{d}}(x,y)\le d(x,y)\) for any \(x,y\in X\). In particular, \(0\le {\bar{d}}(x,x)\le d(x,x)=0\) and then \({\bar{d}}(x,x)=0\) for any \(x\in X\). Therefore, \({\bar{d}}\) is a premetric. Furthermore, \(B_d(x,r)\subseteq B_{{\bar{d}}}(x,r)\subseteq \overline{B_d[x,r]}\) for any \(x\in X\) and \(r\ge 0\). Thus, \({{\mathcal {T}}}_{{\bar{d}}}\subseteq {{\mathcal {T}}}_d\). To prove the inverse inclusion fix \(G\in {{\mathcal {T}}}_d\) and \(x\in G\). By \(T_3\) there is an open set \(U\ni x\) such that \({\overline{U}}\subseteq G\). Since U is a d-neighborhood of x, there is \(\varepsilon >0\) with \(B_d(x,\varepsilon )\subseteq U\). Let \(r=\frac{\varepsilon }{2}\). Then \(B_{{\bar{d}}}(x,r)\subseteq \overline{B_d[x,r]}\subseteq {\overline{U}}\subseteq G\). Thus, G is a \({\bar{d}}\)-neighborhood of x, and so \(G\in {{\mathcal {T}}}_{{\bar{d}}}\). \(\square \)

Theorem 7.2

Let X be a topological space, Y be a metrizable space and Z be a hemimetrizable \(T_3\)-space. Let \(g: X\rightarrow Y\) be a continuous function and \({f: X\times Y\rightarrow Z}\) be a separately continuous function such that the function \({h(x)=f\big (x,g(x)\big )}\), \(x\in X\), is continuous. Then the set \(C_g(f)=\Bigl \{x\in X: f \text { is continuous at }\big (x,g(x)\big ) \Bigr \}\) is residual.

Proof

Let d be a hemimetric which generates the topology of Z, \({\bar{d}}\) be such as in Lemma 7.1, \(\varrho \) be a metric which generates the topology of Y and \({V_m(x)=B_\varrho \big (g(x),\frac{1}{m}\big )}\) for any \(m\in {{\mathbb {N}}}\) and \(x\in X\). Therefore, the premetric \({\bar{d}}\) generates the topology of Z. We will write shortly \({{\widetilde{\omega }}}_f\) instead of \({{\widetilde{\omega }}}_{d,f}\) and \({\bar{\omega }}_f\) instead of \({{\widetilde{\omega }}}_{{\bar{d}},f}\).

Let us prove that \(E=X\setminus C_g(f)\) is meager. Assume, on the contrary, that E is not meager. Set

$$\begin{aligned} E_n=\big \{x\in E:{\bar{\omega }}_f\big (x,g(x)\big )\ge \textstyle \frac{3}{n}\big \}\text { for any }n\in {{\mathbb {N}}}. \end{aligned}$$

By Proposition 6.1(i) since \({\bar{d}}\) generates the topology of Z, we obtain \(E=\bigcup \nolimits _{n=1}^\infty E_n\). Then there exists \(n\in {{\mathbb {N}}}\) such that \(E_n\) is not meager. Set for any \(m\in {\mathbb {N}}\)

$$\begin{aligned} E_{n,m}=\big \{x\in E_n:{{\widetilde{\omega }}}_{f^x}\big (g(x),V_m(x)\big )<\textstyle \frac{1}{n}\big \} \end{aligned}$$

But \(f^x\) is continuous for each \(x\in X\). Consequently, by Proposition 6.1(ii) we obtain \(E_n=\bigcup \nolimits _{m=1}^\infty E_{n,m}\). Since \(E_n\) is not meager, there exists \(m\in {{\mathbb {N}}}\) such that \(E_{n,m}\) is not nowhere dense in X. Hence, there is an open nonempty set \(U_0\subseteq \overline{E}_{n,m}\). Pick an arbitrary point \(a\in U_0\cap E_{n,m}\). Set \(b=g(a)\), \(c=h(a)=f(a,b)\) and \(V=B_\varrho \big (b,\frac{1}{2m}\big )\). By Proposition 6.1(ii), since g and h are continuous at a, there exist an open neighborhood U of a such that \(U\subseteq U_0\) and \(g(U)\subseteq V\) and \({{\widetilde{\omega }}}_{h}(a,U)<\frac{1}{n}\). Therefore, \(\varrho \big (b,g(x)\big )\le \frac{1}{2\,m}\) for any \(x\in U\). Then \(V\subseteq V_m(x)\) for every \(x\in U\). Obviously, the set \(A=U\cap E_{n,m}\) is dense in U. Let \(W=\overline{B_d\big [c,\frac{2}{n}\big ]}\). We prove that \(f(U\times V)\subseteq W\). Fix \(y\in V\). Consider \(x\in A\). Then \(g(x)\in V\subseteq V_m(x)\). Therefore, \({{\widetilde{\omega }}}_{f^x}\big (g(x),V\big )\le {{\widetilde{\omega }}}_{f^x}\big (g(x),V_m(x)\big )<\frac{1}{n}\). Consequently,

$$\begin{aligned} d\big (c,f_y(x)\big )\le d\big (c,f(x,g(x))\big )+d\big (f(x,g(x)),f(x,y)\big )= \\ =d\big (h(a), h(x)\big )+d\big (f^x(g(x)),f^x(y)\big ) \textstyle \le {{\widetilde{\omega }}}_{h}(a,U)+{{\widetilde{\omega }}}_{f^x}(g(x),V)\le \frac{1}{n}+\frac{1}{n}=\frac{2}{n}. \end{aligned}$$

Hence, \(f_y(A)\subseteq W\). But \(f_y\) is a continuous function and \(U\subseteq \overline{A}\). Thus,

$$\begin{aligned} f_y(U)\subseteq f_y\big (\overline{A}\big )\subseteq \overline{f_y(A)}\subseteq \overline{W}=W. \end{aligned}$$

Therefore, \(f(x,y)=f_y(x)\in f_y(U)\subseteq W\) for any \(x\in U\). So, we have proved \(f(U\times V)\subseteq W=\overline{B_d\big [c,\frac{2}{n}\big ]}\). But \({\bar{d}}\big (c,z\big )=\inf \Big \{r\ge 0:z\in \overline{B_d\big [c,r\big ]}\Big \}\le \frac{2}{n}\) for any \(z\in W\). Therefore, \({\bar{d}}\big (f(a,b),f(x,y)\big )\le \frac{2}{n}\) for any \((x,y)\in U\times V\). Since \(a\in E_{n,m}\subseteq E_n\), we conclude that

$$\begin{aligned} \tfrac{3}{n}\le {\bar{\omega }}_f(a,b)\le {\bar{\omega }}_f\big ((a,b),U\times V_m\big )=\sup _{(x,y)\in U\times V_n}{\bar{d}}\big (f(a,b), f(x,y)\big )\le \tfrac{2}{n}, \end{aligned}$$

which is impossible. \(\square \)

Applying Theorem 7.2 for the function \(g(x)=b\), \(x\in X\), we obtain the following generalization of the Baire horizontal theorem.

Theorem 7.3

Let X be a topological space, Y be a first countable space at a point \(b\in Y\), Z be a hemimetrizable \(T_3\)-space and \(f:X\times Y\rightarrow Z\) be a separately continuous function. Then for any \(b\in Y\) the set \(C_b(f)=\Bigl \{x\in X:f\text { is continuous at }(x,b)\Bigr \}\) is residual.

The following example shows that Theorem 7.3 need not be true if Z is a hemimetrizable space without \(T_3\).

Example 7.4

Let \(X=Y=[0;1]\), \(Z=\{0,1\}_0\) be the connected doubleton (Example 3.8) and \({b=0}\). Then there exists a separately continuous function \(f:X\times Y\rightarrow Z\) such that \({C_b(f)=\emptyset }\).

Proof

First of all, we consider the countable set \(A=[0;1]\cap {{\mathbb {Q}}}\) and choose a bijection \({{{\mathbb {N}}}\ni n\mapsto a_n\in A}\). Define for any \((x,y)\in X\times Y\)

$$\begin{aligned} f(x,y)=\left\{ \begin{array}{crl} 0&{},&{}\text { if }x=a_n\text { and }y=\frac{1}{n}\text { for some }n\in {{\mathbb {N}}},\\ 1&{},&{}\text { otherwithe}. \end{array} \right. \end{aligned}$$

Firstly, we prove that \({C_b(f)=\emptyset }\). Fix \(a\in X\). Then \(f(a,b)=f(a,0)=1\) is an isolated point of Z. Thus, \(W_0=\{1\}\) is a neighborhood of f(a, b). Consider neighborhoods U of a and V of b. Since \(\frac{1}{n}\rightarrow 0=b\), there exists \(n_0\in {{\mathbb {N}}}\) such that \(\frac{1}{n}\in V\) for any \(n\ge n_0\). But the set \(\{a_n:n\ge n_0\}\) is dense in X. Consequently, there is \(n\ge n_0\) with \(a_n\in U\). Therefore, \(\big (a_n,\frac{1}{n}\big )\in U\times V\) and \(f\big (a_n,\frac{1}{n}\big )=0\not \in W_0\). Thus, f is discontinuous at (a, b) for any \(a\in X\). So, \({C_b(f)=\emptyset }\).

It remains to prove that f is a separately continuous function. For any \(u\in [0;1]\) we define \(g_u:[0;1]\rightarrow \{0,1\}_0\) by the formula

$$\begin{aligned} g_u(x)=\left\{ \begin{array}{crl} 0&{},&{}\text { if }x=u\,\\ 1&{},&{}\text { otherwise}, \end{array} \right. \ \ \ \text {for any }{x\in [0;1].} \end{aligned}$$

Obviously, \(g_u\) is continuous. But \(f^{a_n}=g_{\frac{1}{n}}\), \(f_{\frac{1}{n}}=g_{a_n}\) and \(f^x=f_y=0\) for any \(x\in [0;1]{\setminus } A\) and \(y\in [0;1]{\setminus }\big \{\frac{1}{n}:n\in {{\mathbb {N}}}\big \}\). So, the functions \(f^x\) and \(f_y\) are continuous for any \(x,y\in [0;1]\). \(\square \)

The following example shows that Theorem 7.2 is not true without the assumption of the continuity of h.

Example 7.5

Let \(X=Y=C\) be the Cantor set and \(Z={{\mathbb {L}}}\) be the Sorgenfrey line (Example 3.6). Then there exists a separately continuous function \(f:X\times Y\rightarrow Z\) such that the diagonal function \(h(x)=f(x,x)\) is discontinuous at every point \(x\in C\setminus C_0\), where \(C_0\) is a countable set.

Proof

It is well known that the map

$$\begin{aligned} q:\{0,1\}^{{\mathbb {N}}}\ni (t_n)_{n=1}^\infty \mapsto \sum \limits _{n=1}^\infty \tfrac{2t_n}{3^n}\in C \end{aligned}$$

is a homeomorphism. Let \(p=q^{-1}:C\rightarrow \{0,1\}^{{\mathbb {N}}}\). Then there are functions \({p_n:C\rightarrow \{0,1\}}\), such that \(p(x)=\big (p_n(x)\big )_{n=1}^\infty \) for any \(x\in C\). Thus, for any \(x\in C\) the following properties hold

$$\begin{aligned}{} & {} x=q(p(x))=\sum \limits _{n=1}^\infty \tfrac{2}{3^n}p_n(x); \end{aligned}$$
(1)
$$\begin{aligned}{} & {} x=y\Leftrightarrow \forall n\in {{\mathbb {N}}}\ |\ p_n(x)=p_n(y); \end{aligned}$$
(2)
$$\begin{aligned}{} & {} \hbox { the sets }U_n(x)=\Big \{y\in C:\forall k\le n\ |\ p_k(x)=p_k(y)\Big \}\text { form a base of} x \hbox { in } C;\nonumber \\ \end{aligned}$$
(3)

Let \(x,y\in C\). Set

$$\begin{aligned} \nu (x,y)=\left\{ \begin{array}{ccl} \min \{n\in {{\mathbb {N}}}:p_n(x)\ne p_n(y)\}&{}, &{} x\ne y; \\ \infty &{}, &{} x=y. \end{array} \right. \end{aligned}$$

Observe, that

$$\begin{aligned} p_k(x)=p_k(y)\text { for any }k<\nu (x,y). \end{aligned}$$
(4)

Let us define \(f:C\times C\rightarrow {{\mathbb {L}}}\) by the formula

$$\begin{aligned} f(x,y)=-\sum \limits _{k<\nu (x,y)}\tfrac{2}{3^k}p_k(x)=-\sum \limits _{k<\nu (x,y)}\tfrac{2}{3^k}p_k(y),\ \ x,y\in C. \end{aligned}$$
(5)

Since \(\nu (x,x)=\infty \), by (5) and (1) we conclude

$$\begin{aligned} h(x)=f(x,x)=-\sum \limits _{k<\nu (x,x)}\tfrac{2}{3^k}p_k(x)=-\sum \limits _{k=1}^\infty \tfrac{2}{3^k}p_k(x)=-x \end{aligned}$$

for all \(x\in C\). By Example 6.2, the function \(h(x)=-x\) is discontinuous at every point \(x\in C\setminus C_0\), where \(C_0\) is a countable set.

Let us prove, that f is separately continuous. Since \(f(x,y)=f(y,x)\) for all \(x,y\in C\), it is sufficient to show, that f is continuous with respect to the first variable. Fix \(x_0,y_0\in C\).

First of all we consider the case \(x_0\ne y_0\). Then \(n=\nu (x_0,y_0)\in {{\mathbb {N}}}\). Let \(x\in U_n(x_0)\). Then \(p_k(x)=p_k(x_0)=p_k(y_0)\) for all \(k<n\) and \(p_n(x)=p_n(x_0)\ne p_n(y_0)\). So, \(\nu (x,y_0)=\nu (x_0,y_0)=n\). Therefore,

$$\begin{aligned} f_{y_0}(x)=f(x,y_0)=-\sum \limits _{k<n}\tfrac{2}{3^k}p_k(x)=f(x_0,y_0)=f_{y_0}(x_0), \end{aligned}$$

for any \(x\in U_n(x_0)\). Thus, \(f_{y_0}\) is constant on \(U_n(x_0)\) and then it is continuous at \(x_0\).

Now we consider the case \(x_0=y_0\). Then \(f_{y_0}(x_0)=-y_0\). Let \(\varepsilon >0\). Choose \(n\in {{\mathbb {N}}}\) with \(\tfrac{1}{3^{n-1}}<\varepsilon \). Fix \(x\in U_n(x_0)\). Then

$$\begin{aligned} f_{y_0}(x)=-\sum \limits _{k<\nu (x,y_0)}\tfrac{2}{3^k}p_k(y_0)\ge -\sum \limits _{k=1}^\infty \tfrac{2}{3^k}p_k(y_0)=-y_0=f_{y_0}(x_0) \end{aligned}$$

On the other hand, since \(\nu (x,y_0)=\nu (x,x_0)>n\), we conclude that

$$\begin{aligned} f_{y_0}(x)= & {} -\sum \limits _{k<\nu (x,y_0)}\tfrac{2}{3^k}p_k(y_0)\le -\sum \limits _{k<n}\tfrac{2}{3^k}p_k(y_0)\\= & {} -\sum \limits _{k=1}^\infty \tfrac{2}{3^k}p_k(y_0)+\sum \limits _{k=n}^\infty \tfrac{2}{3^k}p_k(y_0)\le \le -y_0+\sum \limits _{k=n}^\infty \tfrac{2}{3^k}=f_{y_0}(x_0)\\ {}{} & {} \quad +\tfrac{1}{3^{n-1}}<f_{y_0}(x_0)+\varepsilon . \end{aligned}$$

Thus, \(f_{y_0}\big (U_n(x_0)\big )\subseteq \big [f_{y_0}(x_0);f_{y_0}(x_0)+\varepsilon \big )\). Therefore, \(f_{y_0}\) is continuous at \(x_0\). \(\square \)

8 Choquet, Christensen and Saint-Raymond games

Let us describe the following games (see [8, 9]):

  • the Christensen \(\tau \)-game \(\textrm{Chr}_\tau (X)\);

  • the Christensen \(\sigma \)-game \(\textrm{Chr}_\sigma (X)\);

  • the Christensen game \(\textrm{Chr}(X)\);

  • the Christensen s-game \(\textrm{Chr}_s(X)\);

  • the Saint-Raymond \(\tau \)-game \(\textrm{SR}_\tau (X)\);

  • the Saint-Raymond \(\sigma \)-game \(\textrm{SR}_\sigma (X)\);

  • the Saint-Raymond game \(\textrm{SR}(X)\);

  • the Saint-Raymond s-game \(\textrm{SR}_s(X)\);

  • the Choquet game \(\textrm{Ch}(X)\);

Let \(\Gamma \) be one of these games. In the game \(\Gamma \) there are two players \(\alpha \) and \(\beta \) which move consequently (\(\beta \) starts) choosing \(\alpha _n\) and \(\beta _n\) by the rule of the game \(\Gamma \) (see the tabular below). As a result we obtain a play \(p=(\alpha _n,\beta _n)_{n=1}^\infty \) in \(\Gamma \). If the winning conditions of \(\alpha \) holds then we say that \(\alpha \) wins in the play p in \(\Gamma \). Otherwise, we say that \(\beta \) wins. In the case where \(\alpha \) (or \(\beta \)) has a winning strategy in \(\Gamma \), we say that the game \(\Gamma \) is \(\alpha \)-favorable (\(\beta \)-favorable). In the opposite case we say that \(\Gamma \) is \(\alpha \)-unfavorable (\(\beta \)-unfavorable). In such situations we also say that the space X is \(\alpha \)-(un)favorable (\(\beta \)-(un)favorable) for the game \(\Gamma \).

In the tabular below \(U_n\) and \(V_n\) are non-empty open subset of X.

The game

Moves of the players

The rules of the game

The winning conditions of \(\alpha \)

\(\textrm{Chr}_\tau (X)\)

\(\begin{array}{c} \alpha _n=(U_n,x_n) \\ \beta _n=V_n \end{array}\)

\(\begin{array}{c}V_{n+1}\subseteq U_n\subseteq V_n\\ x_n\in U_n \end{array}\)

Every subnet \((x_{n_k})_{k\in K}\) has a limit point in \(\bigcap \limits _{n\in {{\mathbb {N}}}}U_n\)

\(\textrm{Chr}_\sigma (X)\)

\(\begin{array}{c} \alpha _n=(U_n,x_n) \\ \beta _n=V_n \end{array}\)

\(\begin{array}{c}V_{n+1}\subseteq U_n\subseteq V_n \\ x_n\in U_n \end{array}\)

Every subsequence \((x_{n_k})_{k\in {{\mathbb {N}}}}\) has a limit point in \(\bigcap \limits _{n\in {{\mathbb {N}}}}U_n\)

\(\textrm{Chr}(X)\)

\(\begin{array}{c} \alpha _n=(U_n,x_n) \\ \beta _n=V_n \end{array}\)

\(\begin{array}{c}V_{n+1}\subseteq U_n\subseteq V_n \\ x_n\in U_n \end{array}\)

\((x_n)_{n\in {{\mathbb {N}}}}\) has a limit point in \(\bigcap \limits _{n\in {{\mathbb {N}}}}U_n\)

\(\textrm{Chr}_s(X)\)

\(\begin{array}{c} \alpha _n=(U_n,x_n) \\ \beta _n=V_n \end{array}\)

\(\begin{array}{c}V_{n+1}\subseteq U_n\subseteq V_n\\ x_n\in U_n \end{array}\)

\(\overline{\{x_n:n\in {{\mathbb {N}}}\}}\cap \bigcap \limits _{n\in {{\mathbb {N}}}}U_n\ne \varnothing \)

\(\textrm{SR}_\tau (X)\)

\(\begin{array}{c} \alpha _n=(U_n,x_n) \\ \beta _n=V_n \end{array}\)

\(V_{n+1}\subseteq U_n\subseteq V_n \)

Every subnet \((x_{n_k})_{k\in K}\) has a limit point in \(\bigcap \limits _{n\in {{\mathbb {N}}}}U_n\)

\(\textrm{SR}_\sigma (X)\)

\(\begin{array}{c} \alpha _n=(U_n,x_n) \\ \beta _n=V_n \end{array}\)

\(V_{n+1}\subseteq U_n\subseteq V_n \)

Every subsequence \((x_{n_k})_{k\in {{\mathbb {N}}}}\) has a limit point in \(\bigcap \limits _{n\in {{\mathbb {N}}}}U_n\)

\(\textrm{SR}(X)\)

\(\begin{array}{c} \alpha _n=(U_n,x_n) \\ \beta _n=V_n \end{array}\)

\(V_{n+1}\subseteq U_n\subseteq V_n\)

\((x_n)_{n\in {{\mathbb {N}}}}\) has a limit point in \(\bigcap \limits _{n\in {{\mathbb {N}}}}U_n\)

\(\textrm{SR}_{s}(X)\)

\(\begin{array}{c} \alpha _n=(U_n,x_n) \\ \beta _n=V_n \end{array}\)

\(V_{n+1}\subseteq U_n\subseteq V_n\)

\(\overline{\{x_n:n\in {{\mathbb {N}}}\}}\cap \bigcap \limits _{n\in {{\mathbb {N}}}}U_n\ne \varnothing \)

\(\textrm{Ch}(X)\)

\(\begin{array}{c} \alpha _n=U_n \\ \beta _n=V_n \end{array}\)

\(V_{n+1}\subseteq U_n\subseteq V_n\)

\(\bigcap \limits _{n\in {{\mathbb {N}}}}U_n\ne \varnothing \)

Note, that weaker winning conditions for \(\alpha \) give wider classes of \(\alpha \)-favorable and \(\beta \)-unfavorable spaces. Let \(\Gamma _1\) and \(\Gamma _2\) be two of these games. We write \(\Gamma _1\Rightarrow \Gamma _2\) if \(\alpha \)-favorability of \(\Gamma _1\) implies \(\alpha \)-favorability of \(\Gamma _2\) and \(\beta \)-unfavorability of \(\Gamma _1\) implies \(\beta \)-unfavorability of \(\Gamma _2\). Observe that for any topological space X we have that

$$\begin{aligned} \begin{array}{ccccccccc} \textrm{Chr}_\tau (X)&{}\Rightarrow &{}\textrm{Chr}_\sigma (X)&{}\Rightarrow &{}\textrm{Chr}(X)&{}\Rightarrow &{}\textrm{Chr}_s(X)\\ \Downarrow &{} &{}\Downarrow &{} &{}\Downarrow &{} &{}\Downarrow \\ \textrm{SR}_\tau (X)&{}\Rightarrow &{}\textrm{SR}_\sigma (X)&{}\Rightarrow &{}\textrm{SR}(X)&{}\Rightarrow &{}\textrm{SR}_s(X)&{}\Rightarrow &{}\textrm{Ch}(X)\\ \end{array} \end{aligned}$$

Remark, that every regular (countable) Čech complete space is \(\alpha \)-favorable for the Christensen \(\tau \)-game (\(\sigma \)-game), and \(\beta \)-unfavorability for the Choquet game is equivalent to the Baireness (see [8, 9]). The following example shows that \(\textrm{SR}_\tau (X)\not \Rightarrow \textrm{Chr}_s(X)\).

Example 8.1

Let \({{\mathbb {L}}}\) be the Sorgenfrey line (Example 3.6). Then \({{\mathbb {L}}}\) is \(\alpha \)-favorable for the Saint-Raymond \(\tau \)-game \(\textrm{SR}_\tau ({{\mathbb {L}}})\) but it is \(\beta \)-favorable for the Christensen game \(\textrm{Chr}_s({{\mathbb {L}}})\).

Proof

Firstly, we prove that \({{\mathbb {L}}}\) is \(\alpha \)-favorable for the Saint-Raymond \(\tau \)-game \(\textrm{SR}_\tau ({{\mathbb {L}}})\). Let \((V_k)_{k=1}^n\) be the first n moves of \(\beta \). Choose \(y_n<x_n\) such that \([y_n;x_n]\subseteq V_n\) and \(x_n-y_n<\frac{1}{n}\). Put \(U_n=[y_n;x_n)\) and

$$\begin{aligned} \sigma \bigl ((V_k)_{k=1}^n\bigr )=(U_n,x_n). \end{aligned}$$

Let us prove that \(\sigma \) is a winning strategy for \(\alpha \) in \(\textrm{SR}_\tau ({{\mathbb {L}}})\). Consider a play \(\bigl (V_n,(U_n,x_n)\bigr )_{n=1}^\infty \) in \(\textrm{SR}_\tau (X)\) in which \(\alpha \) plays according to \(\sigma \). Then

$$\begin{aligned}{}[y_{n+1};x_{n+1}]\subseteq V_{n+1}\subseteq U_{n}=[y_n;x_n)\subseteq [y_n;x_n] \end{aligned}$$

and \(x_n-y_n<\frac{1}{n}\) for any \(n\in {{\mathbb {N}}}\). So, there exists a point \(a\in \bigcap \nolimits _{n=1}^\infty [y_n;x_n]\). Hence, \(a\in [y_{n+1};x_{n+1}]\subseteq V_{n+1}\subseteq U_{n}=[y_n;x_n)\) for any \(n\in {{\mathbb {N}}}\). Therefore, \(a\in \bigcap \nolimits _{n=1}^\infty U_n\). On the other hand \(y_{n}\le y_{n+1}\le a\le x_{n+1}\le x_{n}\). Thus, \((x_n)_{n=1}^\infty \) decreases. But \(y_n\le a_n\le x_n<y_n+\frac{1}{n}\le a+\frac{1}{n}\). Then \(x_n\searrow a\). So, \(x_n\rightarrow a\) in \({{\mathbb {L}}}\). Therefore, for any subnet \((x_{n_k})_{k\in K}\) of \((x_n)\) we have that \(x_{n_k}\rightarrow a\) in \({{\mathbb {L}}}\). Hence, \(\alpha \) wins in \(\bigl (V_n,(U_n,x_n)\bigr )_{n=1}^\infty \).

Now we prove that \(\beta \) has a wining strategy in the Christensen s-game \(\textrm{Chr}_s({{\mathbb {L}}})\). Let

$$\begin{aligned} \sigma (\varnothing )=V_1=[a_1;b_1)=[0;1) \end{aligned}$$

be the first move of \(\beta \) in \(\textrm{Chr}_s({{\mathbb {L}}})\). Suppose that \(n\in {{\mathbb {N}}}\) and \((U_1,x_1)\), \((U_2,x_2)\), ..., \((U_n,x_n)\) are the first n moves of \(\alpha \). Since \(x_n\in U_n\), \(U_n\) is a neighborhood of \(x_n\) in \({{\mathbb {L}}}\). Hence, there is \(b_{n+1}>x_n\) such that \([x_n;b_{n+1})\subseteq U_n\). Choose \(a_{n+1}\in (x_n;b_{n+1})\) and set

$$\begin{aligned} \sigma \bigl ((U_k,x_k)_{k=1}^n\bigr )=V_{n+1}=[a_{n+1};b_{n+1}). \end{aligned}$$

Let us prove that \(\sigma \) is a winning strategy for \(\beta \) in \(\textrm{Chr}_s({{\mathbb {L}}})\). Consider a play \(\bigl (V_n,(U_n,x_n)\bigr )_{n=1}^\infty \) in \(\textrm{Chr}_s({{\mathbb {L}}})\) in which \(\beta \) plays according to \(\sigma \). Thus, for any \(n\in {{\mathbb {N}}}\)

$$\begin{aligned} x_{n+1}\in U_{n+1}\subseteq V_{n+1}=[a_{n+1};b_{n+1}) \end{aligned}$$

and then \(x_n<a_{n+1}\le x_{n+1}\). Therefore, \((x_n)_{n=1}^\infty \) increases and \(x_n\not \in V_{n+1}\). So,

$$\begin{aligned} \overline{\{x_n:n\in {{\mathbb {N}}}\}}=\{x_n:n\in {{\mathbb {N}}}\}\subseteq {{\mathbb {L}}}\setminus \bigcap \limits _{n=1}^\infty V_n={{\mathbb {L}}}\setminus \bigcap \limits _{n=1}^\infty U_n. \end{aligned}$$

Thus, \(\beta \) wins in \(\bigl (V_n,(U_n,x_n)\bigr )_{n=1}^\infty \). \(\square \)

9 Generalization of the Kenderov theorem

Theorem 9.1

Let X be a \(\beta \)-unfavorable space for the Christensen s-game \(\textrm{Chr}_s(X)\), (Y, d) be a hemimetric space, \(d'(x,y)=d(y,x)\) for any \(x,y\in Y\) and \(f:X\rightarrow Y\) be a \({{\mathcal {T}}}_d\)-continuous function. Then there exists a dense \(G_\delta \)-subset A of X, such that f is \({{\mathcal {T}}}_{d'}\)-continuous at every point of A. Therefore, the topology \({{\mathcal {T}}}_{d'}\) is categorically related to the topology \({{\mathcal {T}}}_d\) on Y with respect to X.

Proof

Fix \(\varepsilon >0\) and consider the sets

$$\begin{aligned} E_\varepsilon =\Bigl \{x\in X:\forall U\in {{\mathcal {U}}}_x\,\exists x'\in U\,\bigl |\,d\bigl (f(x'),f(x)\bigr )\ge 3\varepsilon \Bigr \} \ \ \ \ \text { and }F_\varepsilon =\overline{E}_\varepsilon , \end{aligned}$$

where \({\mathcal {U}}_x=\Big \{U:U\hbox { is a neighborhood of } x \hbox { in } X\Big \}\). Let us prove that \({\text {int}} F_\varepsilon =\varnothing \). Assuming the contrary let \({V_1={\text {int}} F_\varepsilon \ne \varnothing }\). Now we define some strategy \(\sigma \) for the player \(\beta \) in \(\textrm{Chr}_s(X)\). Let \(\sigma (\varnothing )=V_1\) be the first move of \(\beta \). Consider the first n moves \((U_1,x_1)\), \((U_2,x_2)\), ..., \((U_n,x_n)\) of the player \(\alpha \). Since f is \({{\mathcal {T}}}_d\)-continuous at \(x_n\), there exists an open neighborhood \(W_n\subseteq V_n\) of \(x_n\), such that

$$\begin{aligned} d\bigl (f(x_n),f(x)\bigr )\le \varepsilon \qquad \text { for any }x\in W_n. \end{aligned}$$
(6)

But \(W_n\subseteq V_n\subseteq F_\varepsilon =\overline{E}_\varepsilon \). So, there exists \(y_n\in W_n\cap E_\varepsilon \). Since \(W_n\) is a neighborhood of \(y_n\) and \(y_n\in E_\varepsilon \), there is a point \(z_n\in W_n\) such, that

$$\begin{aligned} d\bigl (f(z_n),f(y_n)\bigr )\ge 3\varepsilon . \end{aligned}$$
(7)

By the \({{\mathcal {T}}}_d\)-continuity of f at \(z_n\), there is an open neighborhood \(V_{n+1}\subseteq W_n\) of \(z_n\), such that

$$\begin{aligned} d\bigl (f(z_n),f(x)\bigr )<\varepsilon \qquad \text { for any }x\in V_{n+1} \end{aligned}$$
(8)

Put \(\sigma \bigl ((U_1,x_1),(U_2,x_2),\dots ,(U_n,x_n)\bigr )=V_{n+1}\).

Since \(\beta \) does not have a winning strategy, \(\sigma \) is not a winning strategy for \(\beta \). Therefore, there exist a play \(\bigl ((U_n,x_n),V_n\bigr )_{n=1}^\infty \) in \(\textrm{Chr}_s(X)\), in which \(\beta \) plays according to its strategy \(\sigma \), but \(\alpha \) wins. Thus, there is a point \(a\in \bigcap \nolimits _{n=1}^\infty U_n\cap \overline{\{x_n:n\in {{\mathbb {N}}}\}}\). Since f is \({\mathcal {T}}_d\)-continuous at a, there exists an neighborhood U of a, such that

$$\begin{aligned} d\bigl (f(a),f(x)\bigr )<\varepsilon \qquad \text { for any }x\in U. \end{aligned}$$
(9)

Since \(a\in \overline{\{x_n:n\in {{\mathbb {N}}}\}}\), choose \(n\in {{\mathbb {N}}}\) with \(x_n\in U\). By (7) and the triangle inequality

$$\begin{aligned} 3\varepsilon{} & {} \le d\bigl (f(z_{n+1}),f(y_{n+1})\bigr )\nonumber \\{} & {} \le d\bigl (f(z_{n+1}),f(a)\bigr )+d\bigl (f(a),f(x_{n})\bigr )+d\bigl (f(x_{n}),f(y_{n+1})\bigr ) \nonumber \\ \end{aligned}$$
(10)

But \(y_{n+1}\in W_{n+1}\subseteq V_{n+1}\subseteq W_n\). So, (6) implies \(d\bigl (f(x_{n}),f(y_{n+1})\bigr )<\varepsilon \). Since \(a\in U_{n+2}\subseteq V_{n+2}\), we have that (8) implies \(d\bigl (f(z_{n+1}),f(a)\bigr )<\varepsilon \). Therefore, we conclude from (10), that \(3\varepsilon < d\bigl (f(a),f(x_{n})\bigr )+2\varepsilon \). Thus,

$$\begin{aligned} d\bigl (f(a),f(x_{n})\bigr )>\varepsilon \end{aligned}$$
(11)

Finally, since \(x_n\in U\), we conclude from (9) and (11), that

$$\begin{aligned} \varepsilon<d\bigl (f(a),f(x_n)\bigr )<\varepsilon . \end{aligned}$$

This contradiction proves that \({\text {int}} F_\varepsilon =\varnothing \).

For any \(n\in {{\mathbb {N}}}\) consider \(G_n=X\setminus F_{\frac{1}{n}}\). Put \(A=\bigcap \nolimits _{n=1}^\infty G_n\). Since \(\overline{G}_n=X\setminus {\text {int}} F_{\frac{1}{n}}=X\) and X is a Baire space, we have that A is a dense \(G_\delta \)-subset of X. Fix \(x_0\in A\) and \(\varepsilon >0\). Choose \(n\in {{\mathbb {N}}}\) with \(\frac{3}{n}<\varepsilon \). Then \(x_0\in A\subseteq G_n=X{\setminus } F_{\frac{1}{n}}\subseteq X{\setminus } E_{\frac{1}{n}}\). So, there exists a neighborhood U of \(x_0\), such that for any \(x\in U\) we have, that \(d'\bigl (f(x_0),f(x)\bigr )=d\bigl (f(x),f(x_0)\bigr )<\frac{3}{n}<\varepsilon \). Therefore, f is \({{\mathcal {T}}}_{d'}\)-continuous at \(x_0\). \(\square \)

Example 9.2

Let \({{\mathbb {L}}}\) be the Sorgenfrey line (Example 3.6) and

$$\begin{aligned} d(x,y)=d'(y,x)=\left\{ \begin{array}{ccl} y-x&{},&{}\text { if }x\le y,\\ 1&{},&{}\text { otherwise}, \end{array} \right. \text { for any }x,y\in {{\mathbb {L}}}. \end{aligned}$$

Then the topology of \({{\mathbb {L}}}\) is generated by d and the identity map \({\text {id}}:{{\mathbb {L}}}\rightarrow {{\mathbb {L}}}\) is d-d-continuous and d-\(d'\)-discontinuous at every point of \({{\mathbb {L}}}.\)

Theorem 9.3

Let X be a \(\beta \)-unfavorable space for the Christensen s-game \(\textrm{Chr}_s(X)\) hemimetrizable \(T_0\)-space (topological space). Then there exists a (pseudo-) metrizable dense \(G_\delta \)-subspace A of X.

Proof

Let d be a hemimetric which generates the topology of X, \(d'(x,y)=d(y,x)\) and \(\widehat{d}(x,y)=\max \{d(x,y),d(y,x)\}\) for any \(x,y\in X\). Consider the identical map \({\text {id}}:X\rightarrow X\). Obviously, \({\text {id}}\) is \({{\mathcal {T}}}_d\)-\({{\mathcal {T}}}_d\)-continuous. Then by Theorem 9.1 we conclude that there is a dense \(G_\delta \)-subset A of X such that \({\text {id}}\) is \({{\mathcal {T}}}_d\)-\({{\mathcal {T}}}_{d'}\)-continuous at every point of A. Thus, by Proposition 3.5 (pseudo-)metric \(\widehat{d}\) generates the topology of A. \(\square \)

Remark that a \(T_0\)-space X from Theorem 9.3 can be non-metrizable. Indeed, both spaces from Examples 3.7 and 3.8 are non-metrizable and have a dense metrizable \(G_\delta \)-subspace \({A=\{1\}}\).

Since every regular Čech complete space is \(\alpha \)-favorable for the Christensen \(\sigma \)-game \(\textrm{Chr}_\sigma (X)\) (see [8]) and then it is a \(\beta \)-unfavorable space for the Christensen s-game \(\textrm{Chr}_s(X)\), Theorem 9.3 implies the following.

Corollary 9.4

Every regular countably Čech complete hemimetrizable space has a metrizable dense \(G_\delta \)-subspace.

10 The Hausdorff hemimetrics and semicontinuity of multifunctions

Definition 9

(Kenderov, [20]) . Let (X, d) be a metric space and the functions \(d_H^-,d_H^+,d_H:2^X\setminus \{\varnothing \}\rightarrow [0;+\infty ]\) are defined by

$$\begin{aligned} d_{H}^-(A,B)= & {} \sup _{x\in A}\inf _{y\in B}d(x,y),\ d_H^+(A,B)=d_H^-(B,A) \\ d_H(A,B)= & {} \max \{d_H^+(A,B),d_H^-(A,B)\} \end{aligned}$$

for any \(A,B\subseteq X.\) We call \(d_H^+\) and \(d_H^-\) the upper and lower Hausdorff hemimetric and \(d_H\) the Hausdorff pseudometric.

So, using notation from Definition 1 we have, that \(d_H^-=(d_H^+)'\) and \({d_H=\widehat{d_H^+}}\).

Definition 10

(Kenderov, [20]). Let X and Y be topological spaces. We write \({F:X\multimap Y}\) instead of \(F:X\rightarrow 2^Y\setminus \{\varnothing \}\) and say that F is a multifunction defined on X and ranged in Y. A multifunction \(F:X\multimap Y\) is called

  • upper (lower) semicontinuous if for any \(a\in X\) and any open set V with \({F(a)\subseteq V}\) (resp. \(F(a)\cap V\ne \varnothing \)) there exists a neighborhood U of a such that \(F(x)\subseteq V\) (resp. \(F(x)\cap V\ne \varnothing \)) for any \(x\in U\).

  • metrically upper (lower) semicontinuous if it is continuous with respect to the upper (lower) Hausdorff hemimetric

It is easy to show that for a compact Y the usual and metrical semicontinuity coincide. Moreover, one can prove the following two propositions.

Proposition 10.1

Let X be a topological space, (Y, d) be a metric space and \({F:X\multimap Y}\) be a compact-valued multifunction. Then F is upper semicontinuous if and only if F is metrically upper semicontinuous

Proposition 10.2

Let X be a topological space, (Y, d) be a metric space and \({F:X\multimap Y}\) be a closed-valued multifunction. Then F is lower semicontinuous if and only if F is metrically lower semicontinuous

Theorem 10.3

(Debs, [11, p.174]). Let X be a topological space, Y be a metrizable space and \(F:X\multimap Y\) be a compact-valued lower continuous multifunction. Then there exists a residual set A in X such that F is upper continuous at every point of A.

Theorem 10.4

(Kenderov, [21, Theorem 5 and Lemma 6]). Let X be a topological space, Y be a second countable space and \(F:X\multimap Y\) be an upper continuous multifunction. Then there exists a residual set A in X such that F is lower continuous at every point of A.

Theorem 10.5

(Kenderov, [20, Theorem 1]). Let X be a Čech complete space, Y be a metric space and \(F:X\multimap Y\) be a metrically upper (lower) semicontinuous multifunction. Then there exists a residual set A in X such that F is metrically lower (upper) semicontinuous at every point of A.

The following corollary of Theorem 9.1 generalizes Theorem 10.5.

Corollary 10.6

Let X be a \(\beta \)-unfavorable space for the Christensen s-game \(\textrm{Chr}_s(X)\), Y be a metric space and \(F:X\multimap Y\) be a metrically upper (lower) semicontinuous multifunction. Then there exists a residual set A in X such that F is metrically lower (upper) semicontinuous at every point of A.

Proof

Since \(F:X\rightarrow 2^Y\) is \(d_H^+\)-continuous (resp. \(d_H^-\)-continuous), Theorem 9.1 implies that there is a dense \(G_\delta \)-set (and hence residual) such that F is continuous on A with respect to the dual hemimetric \(d_H^-=\big (d_H^+\big )'\) (resp. \(d_H^+=\big (d_H^-\big )'\)). Therefore, F is metrically lower (upper) semicontinuous at every point of A. \(\square \)

11 Open problems

It seems that the hemimetric \(d_1\) which is constructed in Examples 5.2 has CBP. So, the following question is very natural.

Problem 1

Does there exist a (completely) regular hemimetrizable space which is not hemimetrizable with CBP?

Obviously, for any premetric space (X, d) the weak triangle inequality \(({\widetilde{\Delta }})\) implies the open ball property

  • (OBP) \(B_d(x,\varepsilon )\) is d-open for any \(x\in X\) and \(\varepsilon >0\); (open ball property)

On the one hand, it seems that the extended premetric \(\varphi \) from the Example 5.2 has (OBP) but does not have \(({\widetilde{\Delta }})\) and then the premetric space \(({{\mathbb {P}}}, \inf \{\varphi ,1\})\) is an example of hemimetrizable premetric space without \(({\widetilde{\Delta }})\). On the other hand, if X is the Alexandroff double circle or the two arrow space (Example 3.2) then we can easily construct a premetric d which generates the topology of X and has (OBP). Therefore, we have the following natural question:

Problem 2

Can we find some simple property (H) of a premetric d (maybe between (OBP) and \(({\widetilde{\Delta }})\)) such that a premetric space (X, d) is hemimetrizable if and only if the property (H) holds?

The following two questions concern generalizations of the Baire theorems.

Problem 3

Let X be a topological space, Y be a metrizable space, Z be a hemimetrizable regular space, \(g:X\rightarrow Y\) be a continuous function and \(f:X\times Y\rightarrow Z\) be a separately continuous function. Under what assumption on Z we can state that there is a residual set A in X such that f is continuous at \(\big (a,g(a)\big )\) for any \(a\in A\)?

Problem 4

Let X be a topological space, Y be a metrizable compact, Z be a hemimetrizable regular space and \(f:X\times Y\rightarrow Z\) be a separately continuous function. Under what assumption on Z we can state that there is a residual set A in X such that f is continuous at \(\big (a,b\big )\) for any \(a\in A\) and \(b\in Y\)?

The Baire curve theorem and the Baire projection theorem (Theorems 1.2  and 1.3) give affirmative answers in the case where Z is metrizable. On the other hand, Example 7.5 show that the Sorgenfrey line is not suitable for Problems 3 and 4.

Despite ours efforts, the following problems are still open.

Problem 5

Let X be a topological space and (Y, d) be an arbitrary hemimetric space. Under what assumption on X we have that every d-continuous function \({f:X\rightarrow Y}\) is \(d'\)-continuous on some residual set A in X?

Theorem 9.1 confirms that the assumption on X must be weaker than the \(\beta \)-unfavorability for the Christensen s-game \(\textrm{Chr}_s(X)\). On the other hand Examples 8.1 and 9.2 show that even the \(\alpha \)-favorability for the Saint Raymond \(\tau \)-game \(\textrm{SR}_{\tau }(X)\) is not enough.

Problem 6

Let X be an arbitrary topological space and (Y, d) be a hemimetric space. Under what assumption on d we have that every d-continuous function \({f:X\rightarrow Y}\) is \(d'\)-continuous on some residual set A in X?

Theorems 10.3 and 10.4 by Debs and Kenderov give affirmative answers for the previous problems in the cases where \((Y,d)=\big (\textrm{Comp}(M),d_H^+\big )\) for a metric space M or \((Y,d)=\big (\textrm{Clsd}(M),d_H^-\big )\) for a separable metric space M (here \(\textrm{Comp}(M)\) means the system of all nonempty compact subsets of M and \(\textrm{Clsd}(M)\) means the system of all nonempty closed subsets of M). Our Theorem 9.1 gives a positive answer for an arbitrary hemimetric d but only for a \(\beta \)-unfavorable space for the Christensen s-game \(\textrm{Chr}_s(X)\). On the other hand, since the Sorgenfrey line \({{\mathbb {L}}}\) is Baire by Example 8.1, we have that Example 9.2 gives a negative answer to Problem 6 for a hemimetric d without any additional assumptions.