On $\alpha$-embedded subsets of products

Abstract

We prove that every continuous function $f:E\rightarrow Y$ depends on countably many coordinates if $E$ is an $(\aleph _1,\aleph _0)$-invariant pseudo-$\aleph _1$-compact subspace of a product of topological spaces and $Y$ is a space with a regular $G_\delta$-diagonal. Using this fact for any $\alpha <\omega _1$, we construct an $(\alpha +1)$-embedded subspace of a completely regular space which is not $\alpha$-embedded.

1 Introduction

If $P$ is a property of functions, then by $P(X)$ ($P^*(X)$) we denote the collection of all real-valued (bounded) functions on a topological space $X$ with the property $P$. By the symbol $C$ we denote the property of continuity. Let $B_\alpha$ be the property of being a function of $\alpha$-th Baire class, where $0\le \alpha <\omega _1$.

Recall that a subset $A$ of $X$ is called functionally closed $($open$)$ in $X$ if there is $f\in C^*(X)$ with $A=f^{-1}(0)$ (). The system of all functionally open (closed) subsets of $X$ we denote by ${\fancyscript{G}}_0^*$ (${{\fancyscript{F}}}_0^*$). Assume that the classes ${\fancyscript{G}}_\xi ^*$ and ${{\fancyscript{F}}}_\xi ^*$ are defined for all $\xi <\alpha$, where $0<\alpha <\omega _1$. Then, if $\alpha$ is odd, the class ${\fancyscript{G}}_\alpha ^*$ (${{\fancyscript{F}}}_\alpha ^*$) consists of all countable intersections (unions) of sets of lower classes, and, if $\alpha$ is even, the class ${\fancyscript{G}}_\alpha ^*$ (${{\fancyscript{F}}}_\alpha ^*$) consists of all countable unions (intersections) of sets of lower classes. The classes ${{\fancyscript{F}}}_\alpha ^*$ for odd $\alpha$ and ${\fancyscript{G}}_\alpha ^*$ for even $\alpha$ are said to be functionally additive, and the classes ${{\fancyscript{F}}}_\alpha ^*$ for even $\alpha$ and ${\fancyscript{G}}_\alpha ^*$ for odd $\alpha$ are called functionally multiplicative. A set $A$ is called functionally measurable if $A\in \bigcup _{0\le \alpha <\omega _1}({{\fancyscript{F}}}_\alpha ^*\cup {\fancyscript{G}}_\alpha ^*)$. Notice that the $\sigma$-algebra of functionally measurable subsets of $X$ is also called the $\sigma$ -algebra of Baire sets.

An important role in the extension theory play $z$-embedded sets (a subset $A$ of a topological space $X$ is called $z$ -embedded in $X$ if for any functionally closed set $F$ in $A$ there exists a functionally closed set $B$ in $X$ such that $B\cap A=F$). In [8] for any $\alpha <\omega _1$ the notion of an $\alpha$ -embedded set was introduced, i.e. such a set $A\subseteq X$ that every its subset $B$ of functionally multiplicative class $\alpha$ in $A$ is the restriction on $A$ of some set of functionally multiplicative class $\alpha$ in $X$. Obviously, the class of $0$-embedded sets coincides with the class of $z$-embedded sets. It is not hard to verify that any $\alpha$-embedded set is $\beta$-embedded if $\alpha \le \beta$ [8, Proposition 2.5]. The converse statement is not true as [8, Theorem 2.6] shows: there is a $1$-embedded subset $E$ of the product , $X_t=\mathbb N$ for all $t\in [0,1]$, which is not $0$-embedded in $X$. Here we generalize this result and show that for any $\alpha <\omega _1$ there exists a set $E\subseteq X$ which is $(\alpha +1)$-embedded but not $\alpha$-embedded in $X$.

A convenient tool in the investigation of properties of $\alpha$-embedded subsets $E$ of $\prod _{t\in T}X_t$ is the fact that under some conditions on $E$ every continuous function $f:E\rightarrow {\mathbb {R}}$ depends on countably many coordinates (see definitions in Sect. 2). Mazur introduced in [10] sets invariant under projection (see Definition 2.1 (a)) and proved that every continuous function $f:E\rightarrow Y$ depends on countably many coordinates if $E\subseteq \mathrm{\Sigma }(a)$ for some $a\in E$ and $E$ is invariant under projection, $X_t$ is a metrizable separable space for each $t\in T$ and $Y$ is a Hausdorff space with a $G_\delta$-diagonal. Engelking [5] established the same result in the case when $E$ is a set which is invariant under composition (see Definition 2.1 (b)) which is contained in $\mathrm{\Sigma }(a)$ for some $a\in E$, $X_t$ is a $T_1$-space with countable base for each $t\in T$ and $Y$ is a Hausdorff space in which every one-point set is $G_\delta$ (see also [7]). Noble and Ulmer [11] obtained the dependence of a continuous function $f:E\rightarrow Y$ on countably many coordinates if $E$ is a subset of a pseudo-$\aleph _1$-compact space $\prod _{t\in T}X_t$, which contains $\sigma (a)$ for some $a\in E$ and $Y$ is a space with a regular $G_\delta$-diagonal. The result of Noble and Ulmer was generalized by Comfort and Gotchev in [2]. Here we consider the so-called $(\aleph _1,\aleph _0)$-invariant subsets of products and, developing the methods of Mazur and of Noble and Ulmer, we show that every continuous function $f:E\rightarrow Y$ depends on countably many coordinates if $E$ is an $(\aleph _1,\aleph _0)$-invariant pseudo-$\aleph _1$-compact subspace of $\prod _{t\in T}X_t$ and $Y$ is a space with a regular $G_\delta$-diagonal.

2 Some properties of pseudo-$\aleph _1$-compact invariant sets

Let $(X_t:t\in T)$ be a family of non-empty topological spaces, $X=\prod _{t\in T}X_t$ and let $a=(a_t)_{t\in T}$ be a fixed point of $X$. For $S\subseteq T$ we denote by $p_S$ the projection $p_S:X\rightarrow \prod _{t\in S}X_t$, where $p_S(x)=(x_t)_{t\in S}$ for each $x=(x_t)_{t\in T}\in X$; by $x_S^a$ we denote the point with coordinates $(y_t)_{t\in T}$, where $y_t=x_t$ if $t\in S$ and $y_t=a_t$ if . For a basic open set $U=\prod _{t\in T}U_t\subseteq X$ let $N(U)=\{t\in T: U_t\ne X_t\}$.

Definition 2.1

A set $E\subseteq X$ is called

• (a) invariant under projection [10] if $x_S^a\in E$ for any $x\in E$ and $S\subseteq T$;

• (b) invariant under composition [5] if for any $x,y\in E$ and $S\subseteq T$ we have $z=(z_t)_{t\in T}\in E$, where $z_t=x_t$ for every $t\in S$ and $z_t=y_t$ for every .

Clearly, every set $E$ invariant under composition is invariant under projection for any $a\in E$.

Following Engelking [5], Hušek in [7, p. 132] introduced a notion of $\kappa$-invariant set for $\kappa \ge \aleph _0$.

Definition 2.2

A set $E$ is called $\kappa$ -invariant if for any $x,y\in E$ and $S\subseteq T$ with $|S|<\kappa$ there is a point $z\in E$ such that $z_t=x_t$ for every $t\in S$ and $z_t=y_t$ for every .

Developing the above-mentioned concepts of Mazur and Hušek, we introduce the following notions.

Definition 2.3

Let $\aleph _i$ and $\aleph _j$ be infinite cardinals, $E\subseteq X$ and $a\in E$. Then $E$ is called

• $\aleph _i$ -invariant with respect to $a$ if $x_S^a\in E$ for every $x\in E$ and $S\subseteq T$ with $|S|<\aleph _i$;

• $(\aleph _i,\aleph _j)$ -invariant with respect to $a$ if $x_{S_1}^a\in E$ and $x_{T\setminus S_2}^a\in E$ for any point $x\in E$ and for any sets $S_1,S_2\subseteq T$ with $|S_1|<\aleph _i$ and .

Obviously, every set $(\aleph _i,\aleph _j)$-invariant with respect to $a$ is $\aleph _i$-invariant with respect to $a$.

Definition 2.4

A topological space $X$ is said to be

• pseudo- $\aleph _1$ -compact if any locally finite family of open subsets of $X$ is at most countable;

• hereditarily pseudo- $\aleph _1$ -compact if each subspace of $X$ is pseudo-$\aleph _1$-compact.

It is easy to check that continuous mappings preserve pseudo-$\aleph _1$-compactness.

The following theorem gives a characterization of pseudo-$\aleph _1$-compactness of $\aleph _0$-invariant sets and is an analogue of the similar result of Noble and Ulmer [11, Corollary 1.5] for products.

Theorem 2.5

Let $(X_t:t\in T)$ be a family of topological spaces, $X=\prod _{t\in T}X_t$, $a\in X$ and let $E\subseteq X$ be an $\aleph _0$-invariant set with respect to $a$. Then the following conditions are equivalent:

• (i) $E$ is pseudo-$\aleph _1$-compact;

• (ii) for any finite non-empty set $S\subseteq T$ and for any uncountable family $(U_i:i\in I)$ of open sets $U_i$ in $X$ with $U_i\cap E\ne \emptyset$ the family $\bigl (p_S(U_i\cap E):i\in I\bigr )$ is not locally finite in $p_S(E)$.

Proof

(i) $\Rightarrow$ (ii)    Let $S\subseteq T$ be a finite non-empty set, $(U_i:i\in I)$ be an uncountable family of basic open sets $U_i$ in $X$ with $U_i\cap E\ne \varnothing$ and let $V_i=p_S(U_i\cap E)$ for each $i\in I$. If the family $(V_i:i\in I)$ is locally finite in $p_S(E)$, then the family $\bigl (p_S^{-1}(V_i)\cap E:i\in I\bigr )$ is locally finite in $E$ and $U_i\cap E\subseteq p_S^{-1}(V_i)\cap E$ for each $i\in I$, which contradicts pseudo-$\aleph _1$-compactness of $E$.

(ii) $\Rightarrow$  (i)   Consider an uncountable family $\bigl (U_i=\prod _{t\in T} U_i^t:i\in I\bigr )$ of basic open sets in $X$ such that $U_i\cap E\ne \varnothing$ for all $i\in I$. By Šanin’s lemma [12] we choose a finite set $Z$ and uncountable set $J\subseteq I$ such that $N(U_i)\cap N(U_j)=Z$ for all distinct $i,j\in J$.

Let $V_i=p_Z(U_i\cap E)$ for all $i\in J$. It follows from (ii) that the family $(V_i:i\in J)$ has a cluster point $v\in p_Z(E)$. Take $y\in E$ such that $v=p_Z(y)$ and put $x=y_Z^a$. We shall show that $x$ is a cluster point of $(U_i\cap E:i\in J)$. Indeed, let $W=\prod _{t\in T}W_t$ be a basic open neighborhood of $x$ in $X$ and $V=\prod _{t\in Z}W_t\cap p_Z(E)$. Choose such infinite set $K\subseteq J$ that $V\cap V_i\ne \varnothing$ and $N(W)\cap N(U_i)\subseteq Z$ for all $i\in K$. Take an arbitrary $i\in K$ and a point $b\in V\cap V_i$. Consider a point $c\in U_i\cap E$ with $b=p_Z(c)$ and put $d=c_{Z\cup N(U_i)}^a$. Clearly, $d\in U_i$. Since $E$ is $\aleph _0$-invariant with respect to $a$ and $c\in E$, $d\in E$. Moreover, $p_Z(d)=p_Z(c)=b\in V$ and $d_t=a_t\in W_t$ for every . Therefore, $d\in W$. Hence, $d\in W\cap E\cap U_i$.$\square$

The example below shows that condition (ii) in the previous theorem cannot be weakened to the following: the set $p_S(E)$ is pseudo- $\aleph _1$ -compact for any non-empty finite set $S\subseteq T$.

Example

There exists a set $E\subseteq \prod _{t\in T}X_t$, $(\aleph _1,\aleph _1)$-invariant with respect to a point $a\in E$ such that $p_S(E)$ is pseudo- $\aleph _1$-compact for any non-empty finite set $S\subseteq T$, but $E$ is not pseudo-$\aleph _1$-compact.

Proof

Let $T=[0,1]$, be the Niemytzki plane [6, p. 21], $X_t=\{0,1\}$ for each $t\in (0,1]$, $X=\prod _{t\in T}X_t$ and let $a=(a_t)_{t\in T}\in X$, where $a_t=0$ for each $t\in (0,1]$ and $a_0=(0,0)$. For each $t\in (0,1]$ define $y^{(t)}=\bigl (y^{(t)}_s\bigr ){}_{s\in T}$ and $z^{(t)}=\bigl (z^{(t)}_s\bigr ){}_{s\in T}\in X$ as follows:

Consider the $(\aleph _1,\aleph _1)$-invariant set

with respect to the point $a$. Observe that for any finite set $S\subseteq [0,1]$ the sets $p_S\bigl (\{y^{(t)}:t\in (0,1]\}\bigr )$ and $p_S\bigl (\{z^{(t)}:t\in (0,1]\}\bigr )$ are finite and the set is separable. Hence, $E$ satisfies the condition mentioned above. But $\bigl (\{y^{(t)}\}:t\in (0,1]\bigr )$ is a locally finite family of open sets in $E$. Therefore, $E$ is not pseudo-$\aleph _1$-compact.

$\square$

3 Dependence of continuous mappings on countably many coordinates

Definition 3.1

Let $E\subseteq \prod _{t\in T}X_t$. We say that a function $f:E\rightarrow Y$ depends on a set $S\subseteq T$ [3, p. 231] if for all $x,y\in E$ the equality $p_S(x)=p_S(y)$ implies $f(x)=f(y)$. If $|S|\le \aleph _0$ then we say that $f$ depends on countably many coordinates. Similarly, $E$ depends on $S$ if for all $x\in E$ and $y\in X$ with $p_S(x)=p_S(y)$ we have $y\in E$.

Definition 3.2

We say that a space $Y$ has a regular $G_\delta$ -diagonal [14] if there exists a sequence $(G_n)_{n=1}^\infty$ of open subsets of $Y^2$ such that

(1)

We denote $\sigma (a)=\bigl \{x\in X:|t\in T:x_t\ne a_t|<\aleph _0\bigr \}$ as in [4].

Theorem 3.3

Let $Y$ be a space with a regular $G_\delta$-diagonal, $(X_t:t\in T)$ be a family of topological spaces, $X=\prod _{t\in T}X_t$, $a\in X$ and let $E\subseteq X$ be a pseudo-$\aleph _1$-compact subspace which is $(\aleph _1,\aleph _0)$-invariant with respect to $a$. Then for any continuous mapping $f:E\rightarrow Y$ there exist a countable set $T_0\subseteq T$ and a continuous mapping $f_0:p_{T_0}(E)\rightarrow Y$ such that . In particular, $f$ depends on countably many coordinates.

Proof

Let $(G_n)_{n=1}^\infty$ be a sequence of open sets in $Y^2$ which satisfies (1) and let $f:E\rightarrow Y$ be a continuous function. Denote by $T_0$ the set of all $t\in T$ for which there exist points $x^t,y^t\in E\cap \sigma (a)$ such that

$$\begin{aligned} x^t_s&=y^t_s \qquad \text {for all}\quad s\ne t \nonumber ,\\ x^t_t&=a_t,\\ f(x^t)&\ne f(y^t).\nonumber \end{aligned}$$

(2)

Assume that $T_0$ is uncountable and choose an uncountable subset $B\subseteq T_0$ and a number $n_0\in \mathbb N$ such that

Using the continuity of $f$ at $x^t$ and $y^t$ for every $t\in B$, we find open basic neighborhoods $U^t$ and $V^t$ of $x^t$ and $y^t$, respectively, such that

(3)

(4)

Since $E$ is pseudo-$\aleph _1$-compact and the family $(V^t\cap E:t\in B)$ is uncountable, there exists a point $x^*\in E$ such that for any basic open neighborhood $W$ of $x^*$ the set $C_W=\{t\in B: V^t\cap E\cap W\ne \varnothing \}$ is infinite. The continuity of $f$ at $x^*$ implies that there is an basic open neighborhood $W$ of $x^*$ such that . Notice that . Fix $t\in C$ and $y\in V^t\cap E\cap W$. Let $x=y^a_{T\setminus \{t\}}$. Then (2) and (3) imply that $x\in U^t$. Since $E$ is $(\aleph _1,\aleph _0)$-invariant with respect to $a$, $x\in E$. Moreover, $x\in W$, since $t\not \in N(W)$. Then $(f(x),f(y))\in G_{n_0}$, which contradicts (4). Hence, the set $T_0$ is countable.

We show that $f$ depends on $T_0$. To do this it is sufficient to check the equality $f(x)=f(x^a_{T_0})$ for every $x\in E$. Consider the case $x\in E\cap \sigma (a)$. Let . Then

$$\begin{aligned} f(x)=f\bigl (x^a_{T\setminus \{t_1\}}\bigr )&= f\bigl (\bigl (x^a_{T\setminus \{t_1\}}\bigr ){}^a_{T\setminus \{t_2\}}\bigr )=\cdots \\&=f\bigl (\bigl (\bigl (x^a_{T\setminus \{t_1\}}\bigr )\cdots \bigr ){}^a_{T\setminus \{t_m\}}\bigr )=f\bigl (x^a_{T_0}\bigr ). \end{aligned}$$

Now let $x\in E$. Notice that $E\cap \sigma (a)$ is a dense set in $E$. Indeed, if $b=(b_t)_{t\in T}\in E$ and $W$ is a basic open neighborhood of $b$ in $X$, then $b^a_{N(W)}\in W\cap E\cap \sigma (a)$. Hence, there exists a net $(x_i)$ of points $x_i\in E\cap \sigma (a)$ such that $\lim _i x_i=x$. Then $\lim _i (x_i)^a_{T_0}=x^a_{T_0}$. It follows from continuity of $f$ that

Consider the function $f_0:p_{T_0}(E)\rightarrow Y$ defined by $f_0(z)=f(x)$ if $z=p_{T_0}(x)$ for $x\in E$. Observe that $f_0$ is defined correctly, because $f$ depends on $T_0$. It remains to prove that $f_0$ is continuous on $p_{T_0}(E)$. Fix $z\in p_{T_0}(E)$ and a net $(z_i)$ of points $z_i\in p_{T_0}(E)$ such that $\lim _i z_i=z$. Take $x\in E$ and $x_i\in E$ with $z=p_{T_0}(x)$ and $z_i=p_{T_0}(x_i)$. Let $y_i=(x_i)^a_{T_0}$ and $y=x^a_{T_0}$. Then $y_i,y\in E$ and $\lim _i y_i=y$. Moreover, since $f$ is continuous at $y$, we have

$$\begin{aligned} \lim \limits _i f_0(z_i)=\lim \limits _i f(x_i)= \lim \limits _i f(y_i)=f(y)=f(x)=f_0(z). \end{aligned}$$

Hence, $f_0$ is continuous at $z$.$\square$

Notice that the proof of dependence of $f$ on $T_0$ in Theorem 3.3 is similar to the proof of [1, Lemmas 2.27 (a) and 2.32].

Theorem 3.4

Let $(X_t:t\in T)$ be an uncountable family of topological spaces, $X=\prod _{t\in T}X_t$, $a\in X$ and let $E\subseteq X$ be an $(\aleph _1,\aleph _0)$-invariant set with respect to $a$. Consider the following conditions:

• (i) $E$ is pseudo-$\aleph _1$-compact;

• (ii) for any space $Y$ with a regular $G_\delta$-diagonal and for any continuous mapping $f:E\rightarrow Y$ there exist a countable set $T_0\subseteq T$ and a continuous mapping $f_0:p_{T_0}(E)\rightarrow Y$ such that ;

• (iii) for any continuous function $f:E\rightarrow {\mathbb {R}}$ there exist a countable set $T_0\subseteq T$ and a continuous mapping $f_0:p_{T_0}(E)\rightarrow {\mathbb {R}}$ such that .

Then (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii).

If $E$ is completely regular and

• (iv) for any non-empty open set $U$ in $E$ there exists an uncountable set $T_U\subseteq T$ such that for every $t\in T_U$ there are $y^{(t)}=\bigl (y^{(t)}_s\bigr ){}_{s\in T}$ and $z^{(t)}=\bigl (z^{(t)}_s\bigr ){}_{s\in T}\in U$ with $y^{(t)}_t\ne z^{(t)}_t$ and $y^{(t)}_s= z^{(t)}_s$ for every ,

then (iii) $\Rightarrow$ (i).

Proof

The implication (i) $\Rightarrow$ (ii) follows from Theorem 3.3, whereas the implication (ii) $\Rightarrow$ (iii) is obvious.

(iii) $\Rightarrow$ (i)   Suppose that $E$ is not pseudo-$\aleph _1$-compact and choose a locally finite in $E$ family $(U_\alpha :\alpha <\omega _1)$ of non-empty open sets $U_\alpha$. Note that $U_\alpha$ may be taken to be disjoint. Indeed, let $(V_i:i\in I)$ be a locally finite family of non-empty open subsets of $E$ with $|I|>\aleph _0$. For every $i\in I$ we choose a non-empty open set $W_i\subseteq V_i$ and a finite set $J_i\subseteq I$ such that $W_i\subseteq \bigcap _{j\in J_i}V_j$ and $W_i\cap V_j=\emptyset$ for all . Since $i\in J_i$ for every $i\in I$, $\bigcup _{i\in I}J_i=I$. Now we take a uncountable set $I_0\subseteq I$ such that all sets $J_i$ from the family $(J_i:i\in I_0)$ are different. Then the uncountable family $(W_i:i\in I_0)$ consists of mutually disjoint elements.

Since $E$ is completely regular, we may assume that all sets $U_\alpha$ are functionally open. For every $\alpha <\omega _1$ take a continuous function $f_\alpha :E\rightarrow [0,1]$ such that $U_\alpha =f_\alpha ^{-1}((0,1])$. Since $T_{U_\alpha }$ is uncountable, we may construct a family $(t_\alpha :\alpha <\omega _1)$ of distinct points $t_\alpha \in T_{U_\alpha }$. According to (iv) we choose for every $\alpha <\omega _1$ points $y^{(\alpha )}=\bigl (y^{(\alpha )}_s\bigr ){}_{s\in T}$, $z^{(\alpha )}=\bigl (z^{(\alpha )}_s\bigr ){}_{s\in T}\in U_\alpha$ such that $y^{(\alpha )}_{t_\alpha }\ne z^{(\alpha )}_{t_\alpha }$ and $y^{(\alpha )}_s= z^{(\alpha )}_s$ for every . Now for every $\alpha <\omega _1$ we choose a continuous function $g_\alpha :E\rightarrow [0,1]$ such that $g_\alpha \bigl (y^{(\alpha )}\bigr )=1$ and $g_\alpha \bigl (z^{(\alpha )}\bigr )=0$.

Consider the continuous function $f:E\rightarrow [0,1]$, . Since sets $U_\alpha$ are mutually disjoint,

Hence, $f\bigl (y^{(\alpha )}\bigr )\ne f\bigl (z^{(\alpha )}\bigr )$ for every $\alpha <\omega _1$. Since the set $\{t_{\alpha }:\alpha <\omega _1\}$ is uncountable, the function $f$ does not satisfy (iii).$\square$

4 Functionally measurable sets

Proposition 4.1

Let $E$ be a subset of $X=\prod _{t\in T}X_t$ such that for any continuous function $f:E\rightarrow {\mathbb {R}}$ there exist a countable set $T_0\subseteq T$ and a continuous mapping $f_0:p_{T_0}(E)\rightarrow {\mathbb {R}}$ with and let $0\le \alpha <\omega _1$. Then for any set $A$ of functionally additive $($multiplicative$)$ class $\alpha$ in $E$ there exists a countable set $T_0\subseteq T$ such that $A$ depends on $T_0$ and $p_{T_0}(A)$ is of functionally additive $($multiplicative$)$ class $\alpha$ in $p_{T_0}(E)$.

Proof

Let $\alpha =0$. We consider the case when a set $A$ is functionally open in $E$. Then $A=f^{-1}((0,+\infty ))$ for some continuous function $f:E\rightarrow {\mathbb {R}}$. Take a countable set $T_0\subseteq T$ and a continuous mapping $f_0:p_{T_0}(E)\rightarrow {\mathbb {R}}$ with . Then the set $p_{T_0}(A)=f_0^{-1}((0,+\infty ))$ is functionally open in $p_{T_0}(E)$. Moreover, if $x\in A$ and $y\in E$ with $p_{T_0}(x)=p_{T_0}(y)$, then $f(y)=f(x)>0$. Therefore, $y\in A$ which implies that $A$ depends on $T_0$.

Assume that the assertion is true for all $\alpha <\beta$ and consider a set $A$ of functionally additive class $\alpha$ in $E$. Then $A=\bigcup _{n=1}^\infty A_n$, where $A_n$ is of functionally multiplicative class $\alpha _n<\alpha$ for every $n$. By the assumption, for every $n$ there exists a countable set $T_n\subseteq T$ such that $A_n$ depends on $T_n$ and $p_{T_n}(A_n)$ belongs to functionally multiplicative class $\alpha _n$ in $p_{T_n}(E)$. Notice that $p_{T_0}(A_n)$ is of functionally multiplicative class $\alpha _n$ in $p_{T_0}(E)$ for every $n$. Then $p_{T_0}(A)=\bigcup _{n=1}^\infty p_{T_0}(A_n)$ is of functionally additive class $\alpha$ in $p_{T_0}(E)$.$\square$

Definition 4.2

Let $0\le \alpha <\omega _1$. A space $X$ is called $\alpha$ -universal if any subset of $X$ is $\alpha$-embedded in $X$.

Clearly, every perfectly normal space is $\alpha$-universal for any $\alpha <\omega _1$.

Proposition 4.3

Let $0\le \alpha <\omega _1$, $(X_t)_{t\in T}$ be a family of topological spaces such that every countable subproduct is $\alpha$-universal, $X=\prod _{t\in T}X_t$ and let $E\subseteq X$ be such a set as in Proposition 4.1. Then $E$ is an $\alpha$-embedded set in $X$.

Proof

Let $A\subseteq E$ be a set of functionally multiplicative class $\alpha$ in $E$. According to Proposition 4.1 there exists a countable set $T_0\subseteq T$ such that $A$ depends on $T_0$ and $A_0=p_{T_0}(A)$ is of functionally multiplicative class $\alpha$ in $E_0=p_{T_0}(E)$. Since $X_0=\prod _{t\in T_0}X_t$ is $\alpha$-universal, the set $E_0$ is $\alpha$-embedded in $X_0$. Hence, there exists a set $B_0$ of functionally multiplicative class $\alpha$ in $X_0$ such that $B_0\cap E_0=A_0$. Let $B=p_{T_0}^{-1}(B_0)$. Then $B$ is of functionally multiplicative class $\alpha$ in $X$, because the mapping $p_{T_0}$ is continuous. Moreover, it is easy to see that $B\cap E=A$.$\square$

Proposition 4.4

Let $0\le \alpha <\omega _1$, $X=\prod _{t\in T}X_t$ be a pseudo-$\aleph _1$-compact space, where $(X_t)_{t\in T}$ is a family of spaces such that every countable subproduct is $\alpha$-universal and hereditarily pseudo-$\aleph _1$-compact. Then any functionally measurable set $E\subseteq X$ is $\alpha$-embedded in $X$.

Proof

Consider a functionally measurable set $E\subseteq X$. Without loss of generality, we may assume that $E$ belongs to functionally multiplicative class $\beta$ for some $0\le \beta <\omega _1$. Take a function $f\in B_\beta (X)$ such that $E=f^{-1}(0)$. Since $X$ is pseudo-$\aleph _1$-compact, [11, Theorem 2.3] implies that there exists a countable set $T_0\subseteq T$ such that for all $x\in E$ and $y\in X$ the equality $p_{T_0}(x)=p_{T_0}(y)$ implies that $y\in E$. Let $E_0=p_{T_0}(E)$. Then

Since $\prod _{t\in T_0\cup S} X_t$ is a hereditarily pseudo-$\aleph _1$-compact space, is pseudo-$\aleph _1$-compact space for any finite set . Hence, by [11, Corollary1.5] the set $E$ is pseudo-$\aleph _1$-compact. Therefore, $E$ satisfy the condition of Proposition 4.1 by Theorem 3.3 applied to the whole product . It remains to use Proposition 4.3.$\square$

The following result implies a positive answer to [8, Question 8.1].

Corollary 4.5

Let $(X_t)_{t\in T}$ be a family of separable metrizable spaces. Then every functionally measurable subset of $X=\prod _{t\in T} X_t$ is $\alpha$-embedded in $X$ for any $0\le \alpha <\omega _1$.

Proof

The statement follows from Proposition 4.4 and the fact that any countable product of separable metrizable spaces is separable and metrizable, consequently, $\alpha$-universal and hereditarily pseudo-$\aleph _1$-compact.$\square$

5 The construction of $\alpha$-embedded sets

Theorem 5.1

For every $0\le \alpha <\omega _1$ there exists a completely regular space $X$ with an $(\alpha +1)$-embedded subspace $E\subseteq X$ which is not $\alpha$-embedded.

Proof

Fix $\alpha <\omega _1$. Let $X_0=[0,1]$, $X_t=\mathbb N$ for every $t\in (0,1]$, $Y=\prod _{t\in (0,1]}X_t$ and .

According to [9, p. 371] there exists a set $A_1\subseteq [0,1]$ of additive class $\alpha$ which does not belong to multiplicative class $\alpha$. Let . For $i=1,2$ put

It is easy to see that $F_1$ and $F_2$ are closed disjoint subsets of $Y$. Let for $i=1,2$ and $E=B_1\cup B_2$. Then $B_1$ and $B_2$ are disjoint closed subsets of $E$.

Claim 5.2

The set $B_i$ is $\alpha$-embedded in $X$ for every $i=1,2$.

Proof

We show that $B_1$ is pseudo-$\aleph _1$-compact (for the set $B_2$ we argue verbatim). Since $A_1$ is separable, it is enough to check that $F_1$ is pseudo-$\aleph _1$-compact. Notice that the set $F_1$ is $(\aleph _1,\aleph _1)$-invariant with respect to the point $a=(a_t)_{t\in (0,1]}$, where $a_t=1$ for every $t\in (0,1]$. Since for any finite set $S\subseteq (0,1]$ the space $\prod _{t\in S}X_t$ is countable, the set $F_1$ satisfies condition (ii) of Theorem 2.5. Then by Theorem 2.5 the set $F_1$ is pseudo-$\aleph _1$-compact.

Now observe that each set $B_i$ is $(\aleph _1,\aleph _1)$-invariant with respect to the point $a^i=(a_t^i)_{t\in [0,1]}$, where $a_t^i=i$ for all $t\in (0,1]$ and $a_0^i\in A_i$. It remains to apply Theorem 3.3 and Proposition 4.3. $\blacksquare$

Claim 5.3

The set $E$ is not $\alpha$-embedded in $X$.

Proof

Assume the contrary and choose a set $H$ of functionally multiplicative class $\alpha$ in $X$ such that $H\cap E=B_1$. It follows from Proposition 4.1 that there is a countable set $S=\{0\}\cup T$, where $T\subseteq (0,1]$, such that $H$ depends on $S$. Let $y_0\in Y$ be such that $p_T(y_0)$ is a sequence of distinct natural numbers which are not equal to $1$ or $2$. Take $y_1\in F_1$ and $y_2\in F_2$ with $p_T(y_0)=p_T(y_1)=p_T(y_2)$. Then for all $x\in A_1$ we have $(x,y_1)\in H$ and, consequently, $(x,y_0)\in H$. Moreover, for all $x\in A_2$ we have $(x,y_2)\not \in H$ and, consequently, $(x,y_0)\not \in H$. Hence, . Therefore, is of functionally multiplicative class $\alpha$ in $X$, which implies that the set $A_1$ belongs to functionally multiplicative class $\alpha$ in $[0,1]$, a contradiction. $\blacksquare$

Claim 5.4

The set $E$ is $(\alpha +1)$-embedded in $X$.

Proof

Let $C$ be a set of functionally multiplicative class $(\alpha +1)$ in $E$. Denote for $i=1,2$. Then $E_1$ is of functionally additive class $\alpha$ and $E_2$ is of functionally multiplicative class $\alpha$ in $X$. For $i=1,2$ put $C_i=C\cap B_i$. Since each of the sets $C_i$ is of functionally multiplicative class $(\alpha +1)$ in the $\alpha$-embedded set $B_i$ in $X$, there exists a set $D_i$ of functionally multiplicative class $(\alpha +1)$ in $X$ such that $D_i\cap B_i=C_i$. Let $D=(D_1\cap E_1)\cup (D_2\cap E_2)$. Then $D$ is a set of functionally multiplicative class $(\alpha +1)$ in $X$ and $D\cap E=C$.$\blacksquare$

This completes the proof.$\square$

Notice that the sets $F_i$ were first defined by Stone [13] in his proof of non-normality of the uncountable power $\mathbb N^\tau$ of the space $\mathbb N$ of natural numbers.