Josefson–Nissenzweig property for \(C_{p}\)spaces
 112 Downloads
Abstract
The famous Rosenthal–Lacey theorem asserts that for each infinite compact space K the Banach space C(K) admits a quotient isomorphic to Banach spaces c or \(\ell _{2}\). The aim of the paper is to study a natural variant of this result for the space \(C_{p}(X)\) of continuous realvalued maps on a Tychonoff space X with the pointwise topology. Following Josefson–Nissenzweig theorem for infinitedimensional Banach spaces we introduce a corresponding property (called Josefson–Nissenzweig property, briefly, the JNP) for \(C_{p}(X)\)spaces. We prove: for a Tychonoff space X the space \(C_p(X)\) satisfies the JNP if and only if \(C_p(X)\) has a quotient isomorphic to \(c_{0}:=\{(x_n)_{n\in \mathbb N}\in \mathbb {R}^\mathbb {N}:x_n\rightarrow 0\}\) (with the product topology of \(\mathbb {R}^\mathbb {N}\)) if and only if \(C_{p}(X)\) contains a complemented subspace isomorphic to \(c_0\). The last statement provides a \(C_{p}\)version of the Cembranos theorem stating that the Banach space C(K) is not a Grothendieck space if and only if C(K) contains a complemented copy of the Banach space \(c_{0}\) with the supnorm topology. For a pseudocompact space X the space \(C_p(X)\) has the JNP if and only if \(C_p(X)\) has a complemented metrizable infinitedimensional subspace. An example of a compact space K without infinite convergent sequences with \(C_{p}(K)\) containing a complemented subspace isomorphic to \(c_{0}\) is given.
Keywords
The separable quotient problem Spaces of continuous functions Quotient spaces The Josefson–Nissenzweig theorem Efimov spaceMathematics Subject Classification
46E10 54C351 Introduction, motivations and two main problems
Let X be a Tychonoff space. By \(C_{p}(X)\) we denote the space of realvalued continuous functions on X endowed with the pointwise topology.
We will need the following fact stating that each metrizable (linear) quotient \(C_{p}(X)/Z\) of \(C_p(X)\) by a closed vector subspace Z of \(C_p(X)\) is separable. Indeed, this follows from the separability of metizable spaces of countable cellularity and the fact that \(C_p(X)\) has countable cellularity, being a dense subspace of \(\mathbb {R}^X\), see [2].
The classic Rosenthal–Lacey theorem, see [19, 23, 27], asserts that the Banach space C(K) of continuous realvalued maps on an infinite compact space K has a quotient isomorphic to Banach spaces c or \(\ell _{2},\) or equivalently, there exists a continuous linear (and open; by the open mapping Banach theorem) map from C(K) onto c or \(\ell _{2}\), see also a survey paper [14].
This theorem motivates the following natural question for spaces \(C_{p}(X)\).
Problem 1
 (1)
The space \(C_{p}(K)\) has an infinite dimensional metrizable quotient.
 (2)
The space \(C_{p}(K)\) has an infinite dimensional metrizable separable quotient.
 (3)
The space \(C_p(K)\) has a quotient isomorphic to a dense subspace of \(\mathbb {R}^{\mathbb {N}}\).
In [21] it was shown that \(C_{p}(K)\) has an infinitedimensional separable quotient algebra if and only if K contains an infinite countable closed subset. Hence \(C_{p}(\beta \mathbb {N})\) lacks infinitedimensional separable quotient algebras. Nevertheless, as proved in [22, Theorem 4], the space \(C_{p}(K)\) has infinitedimensional separable quotient for any compact space K containing a copy of \(\beta \mathbb {N}\).
Problem 1 has been already partially studied in [3], where we proved that for a Tychonoff space X the space \(C_p(X)\) has an infinitedimensional metrizable quotient if X either contains an infinite discrete \(C^{*}\)embedded subspace or else X has a sequence \((K_n)_{n\in \mathbb {N}}\) of compact subsets such that for every n the space \(K_n\) contains two disjoint topological copies of \(K_{n+1}\). If fact, the first case (for example if compact X contains a copy of \(\beta \mathbb {N}\)) asserts that \(C_{p}(X)\) has a quotient isomorphic to the subspace \(\ell _\infty =\{(x_n)\in \mathbb {R}^{\mathbb {N}}:\sup _n x_n<\infty \}\) of \(\mathbb {R}^{\mathbb {N}}\) or to the product \(\mathbb {R}^{\mathbb {N}}\).
Consequently, this theorem reduces Problem 1 to the case when K is an Efimov space (i.e. K is an infinite compact space that contains neither a nontrivial convergent sequence nor a copy of \(\beta \mathbb {N}\)). Although, it is unknown if Efimov spaces exist in ZFC (see [7, 8, 9, 10, 12, 13, 16, 18]) we showed in [22] that under \(\lozenge \) for some Efimov spaces K the function space \(C_{p}(K)\) has an infinite dimensional metrizable quotient.
By \(c_{0}\) we mean the subspace \(\{(x_n)_{n\in \mathbb N}\in \mathbb {R}^{\mathbb {N}}:x_n\rightarrow 0\}\) of \(\mathbb {R}^{\mathbb {N}}\) endowed with the product topology. The term “the Banach space \({c}_{0}\)” means the classic Banach space of nullsequences with the supnorm topology.
It is known that the Banach space C(K) over an infinite compact K contains a copy of the Banach space \(c_{0}\), see for example [6]. By a result of Cembranos, see [4, Theorem, page 74], the space C(K) is not a Grothendieck space if and only if C(K) contains a complemented copy of the Banach space \(c_{0}\). Recall a Banach space E is a Grothendieck space if every weak\(^{*}\) converging sequence in the dual \(E^{*}\) weakly converges in \(E^{*}\). It is wellknown that if a compact space K contains a nontrivial converging sequence, C(K) is not a Grothendieck space; hence C(K) contains a complemented copy of the Banach space \(c_{0}\). It is also easy to see that for every infinite compact space K the space \(C_{p}(K)\) contains a closed copy of the space \(c_{0}\) endowed with the product topology of \(\mathbb {R}^{\mathbb {N}}\).
Cembranos theorem motivates the following next problem (connected with Problem 1).
Problem 2
Characterize those spaces \(C_{p}(K)\) which contain a complemented copy of \(c_{0}\) with the product topology of \(\mathbb {R}^{\mathbb {N}}\).
2 The main results
For a Tychonoff space X and a point \(x\in X\) let \(\delta _x:C_p(X)\rightarrow \mathbb {R},\,\,\, \delta _x:f\mapsto f(x),\) be the Dirac measure concentrated at x. The linear hull \(L_p(X)\) of the set \(\{\delta _x:x\in X\}\) in \(\mathbb {R}^{C_p(X)}\) can be identified with the dual space of \(C_p(X)\). We refer also the reader to [15] for more information about the dual \(L_p(X)\).
Elements of the space \(L_p(X)\) will be called finitely supported signmeasures (or simply signmeasures) on X.
Each \(\mu \in L_p(X)\) can be uniquely written as a linear combination of Dirac measures \(\mu =\sum _{x\in F}\alpha _x\delta _x\) for some finite set \(F\subset X\) and some nonzero real numbers \(\alpha _x\). The set F is called the support of the signmeasure \(\mu \) and is denoted by \(\mathrm {supp}(\mu )\). The measure \(\sum _{x\in F}\alpha _x\delta _x\) will be denoted by \(\mu \) and the real number \(\Vert \mu \Vert =\sum _{x\in F}\alpha _x\) coincides with the norm of \(\mu \) (in the dual Banach space \(C(\beta X)^{*}\)).
The signmeasure \(\mu =\sum _{x\in F}\alpha _x\delta _x\) determines the function \(\mu :2^X\rightarrow \mathbb {R}\) defined on the powerset of X and assigning to each subset \(A\subset X\) the real number \(\sum _{x\in A\cap F}\alpha _x\). So, a finitely supported signmeasure will be considered both as a linear functional on \(C_p(X)\) and an additive function on the powerset \(2^X\).
The famous Josefson–Nissenzweig theorem asserts that for each infinitedimensional Banach space E there exists a null sequence in the weak\(^{*}\)topology of the topological dual \(E^{*}\) of E and which is of norm one in the dual norm, see for example [6].
We propose the following corresponding property for spaces \(C_{p}(X)\).
Definition 1
For a Tychonoff space X the space \(C_{p}(X)\) satisfies the Josefson–Nissenzweig property (JNP in short) if there exists a sequence \((\mu _n)\) of finitely supported signmeasures on X such that \(\Vert \mu _n\Vert =1\) for all \(n\in \mathbb {N}\), and \(\mu _n(f)\rightarrow _n 0\) for each \(f\in C_p(X)\).
 (1)
If a compact spaceKcontains a nontrivial convergent sequence, say\(x_{n}\rightarrow x\), then\(C_{p}(K)\)satisfies the JNP. This is witnessed by the weak\(^{*}\) null sequence \((\mu _n)\) of signmeasures \(\mu _{n}=\frac{1}{2}(\delta _{x_{n}}\delta _{x})\), \(n\in \mathbb {N}\).
 (2)
The space\(C_{p}(\beta \mathbb {N})\)does not satisfy the JNP. This follows directly from the Grothendieck theorem, see [5, Corollary 4.5.8].
 (3)
There exists a compact spaceKcontaining a copy of\(\beta \mathbb {N}\)but without nontrivial convergent sequences such that\(C_{p}(K)\)satisfies the JNP, see Example 1 below.
It turns out that the Josefson–Nissenzweig property characterizes an interesting case related with Problem 1 and provides a complete solution to Problem 2.
Theorem 1
 (1)
\(C_{p}(X)\) satisfies the JNP;
 (2)
\(C_p(X)\) contains a complemented subspace isomorphic to \(c_0\);
 (3)
\(C_p(X)\) has a quotient isomorphic to \(c_0\);
 (4)
\(C_p(X)\) admits a linear continuous map onto \(c_0.\) If the space X is pseudocompact, then the conditions (1)–(4) are equivalent to
 (5)
\(C_{p}(X)\) contains a complemented infinitedimensional metrizable subspace;
 (6)
\(C_{p}(X)\) contains a complemented infinitedimensional separable subspace;
 (7)
\(C_p(X)\) has an infinitedimensional Polishable quotient.
We recall that a locally convex space X is Polishable if X admits a stronger Polish locally convex topology. Equivalently, Polishable locally convex spaces can be defined as images of separable Fréchet spaces under continuous linear maps. Clearly, the subspace \(c_0\) of \(\mathbb {R}^{\mathbb {N}}\) is Polishable.
A topological space X is pseudocompact if it is Tychonoff and each continuous realvalued function on X is bounded. It is known (see [3]) that a Tychonoff space X is not pseudocompact if and only if \(C_{p}(X)\) contains a complemented copy of \(\mathbb {R}^{\mathbb {N}}\). Combining this characterization with Theorem 1, we obtain another characterization related to Problem 1.
Corollary 1
 (1)
\(C_p(X)\) has an infinitedimensional Polishable quotient;
 (2)
\(C_p(X)\) contains a complemeneted infinitedimensional Polishable subspace;
 (3)
\(C_p(X)\) contains a complemented subspace isomorphic to \(\mathbb {R}^{\mathbb {N}}\) or \(c_0\);
Corollary 2
 (1)
has a quotient isomorphic to \(\ell _{\infty }\);
 (2)
contains a subspace isomorphic to \(c_{0}\);
 (3)
does not admit a continuous linear map onto \(c_0\);
 (4)
has no Polishable infinitedimensional quotients;
 (4)
contains no complemented separable infinitedimensional subspaces.
Indeed, the first claim follows from [3, Proposition], the others follow from Theorem 1 and the statement (2) after Definition 1.
In the final Sect. 5 we shall characterize Tychonoff spaces X whose function space \(C_p(X)\) is Polishable and prove the following theorem.
Theorem 2
 (1)
\(C_p(X)\) is Polishable;
 (2)
\(C_k(X)\) is Polishable;
 (3)
\(C_k(X)\) is Polish;
 (4)
X is a submetrizable hemicompact kspace.
In this theorem \(C_k(X)\) denotes the space of continuous realvalued functions on X, endowed with the compactopen topology. It should be mentioned that a locally convex space is Polish if and only if it is a separable Fréchet space, by using, for example, the Birkhoff–Kakutani theorem [20, Theorem 9.1].
3 Proof of Theorem 1
We start with the following
Lemma 1
 (1)
\(\Vert \mu _n\Vert =1\) for all \(n\in \mathbb {N}\), and
 (2)
\(\mu _n(f)\rightarrow _n 0\) for all \(f\in C(K)\).
 (a)
the closed subspace \(Z=\bigcap _{k\in \Omega }\{f\in C_p(X):\mu _k(f)=0\}\) of \(C_p(X)\) is complemented in the subspace \(L=\big \{f\in C_p(X):\lim _{k\in \Omega }\mu _k(f)=0\big \}\) of \(C_p(X)\);
 (b)
the quotient space L / Z is isomorphic to the subspace \(c_0\) of \(\mathbb {R}^\mathbb {N}\);
 (c)
L contains a complemented subspace isomorphic to \(c_0\);
 (d)
the quotient space \(C_p(X)/Z\) is infinitedimensional and metrizable (and so, separable).
Proof
 (I)
First we show that the set \(M=\{\mu _n: n\in \mathbb {N}\}\) is not relatively weakly compact in the dual of the Banach space C(K). Indeed, assume on the contrary that the closure \(\overline{M}\) of M in the weak topology of \(C(K)^{*}\) is weakly compact. Applying the Eberlein–Šmulian theorem [1, Theorem 1.6.3], we conclude that \(\overline{M}\) is weakly sequentially compact. Thus \((\mu _n)\) has a subsequence \((\mu _{k_n})\) that weakly converges to some element \(\mu _0\in C(K)^{*}\). Taking into account that the sequence \((\mu _n)\) converges to zero in the weak\(^{*}\) topology of \(C(K)^{*}\), we conclude that \(\mu _0=0\) and hence \((\mu _{k_n})\) is weakly convergent to zero in \(C(K)^{*}\). Denote by W the countable set \(\bigcup _{n\in \mathbb {N}}\mathrm {supp}(\mu _n)\). The measures \(\mu _n, n\in \mathbb {N},\) can be considered as elements of the unit sphere of the Banach space \(\ell _1(W)\subset C(K)^{*}\). By the Schur theorem [1, Theorem 2.3.6], the weakly convergent sequence \((\mu _{k_n})\) is convergent to zero in the norm topology of \(\ell _1(W)\), which is not possible as \(\Vert \mu _n\Vert =1\) for all \(n\in \mathbb {N}\). Thus the set M is not relatively weakly compact in \(C(K)^{*}\).
 (II)By the Grothendieck theorem [1, Theorem 5.3.2] there exist a number \(\epsilon >0\), a sequence \((m_n) \subset \mathbb {N}\) and a sequence \((U_n)\) of pairwise disjoint open sets in K such that \(\mu _{m_n}(U_n)>\epsilon \) for any \(n\in \mathbb {N}\). Clearly, \(\lim _{n\rightarrow \infty }\mu _k(U_n)=0\) for any \(k\in \mathbb {N}\), sinceThus we can assume that the sequence \((m_n)\) is strictly increasing.$$\begin{aligned} \sum _{n\in \mathbb {N}} \mu _k(U_n) = \mu _k\left( \bigcup _{n\in \mathbb {N}} U_n\right) \le \mu _k(K)=1. \end{aligned}$$
 (A1)
\(\nu _k(f)\rightarrow _k 0\) for every \(f\in C(K);\)
 (A2)
\(\nu _k(W_k)>\epsilon \) for every \(k\in \mathbb {N};\)
 (A3)
\(\nu _k(W_n)=0\) for all \(k,n\in \mathbb {N}\) with \(k<n.\)
 (III)By induction we shall construct a decreasing sequence \((N_k)\) of infinite subsets of \(\mathbb {N}\) with \(\min N_k< \min N_{k+1}\) for \(k\in \mathbb {N}\) such that \(\nu _n(W_m)\le \epsilon /3^k\) for every \(k\in \mathbb {N}, m=\min N_k, n\in N_k\) and \(n>m\). Let \(N_0=\mathbb {N}.\) Assume that for some \(k\in \mathbb {N}\) an infinite subset \(N_{k1}\) of \(\mathbb {N}\) has been constructed. Let F be a finite subset of \(N_{k1}\) with \(F>3^k/\epsilon \) and \(\min F> \min N_{k1}.\) For every \(i\in F\) consider the setFor every \(n\in N_{k1}\) we get \(\nu _n(X)\ge \sum _{i\in F} \nu _n(W_i).\) Hence there exists \(i\in F\) such that$$\begin{aligned} \Lambda _i=\{n\in N_{k1}: \nu _n(W_i) \le \epsilon /3^k\}. \end{aligned}$$Thus \(N_{k1}=\bigcup _{i\in F} \Lambda _i,\) so for some \(m\in F\) the set \(\Lambda _m\) is infinite. Put$$\begin{aligned} \nu _n(W_i)\le 1/F \le \epsilon /3^k. \end{aligned}$$Then \(\min N_{k1}< \min F \le m=\min N_k\) and \(\nu _n(W_m)\le \epsilon /3^k\) for \(n\in N_k\) with \(n>m.\)$$\begin{aligned} N_k=\{n\in \Lambda _m: n>m\}\cup \{m\}. \end{aligned}$$
 (IV)Let \(i_k= \min N_k, \lambda _k=\nu _{i_k}\) and \(V_k=W_{i_k}\) for \(k\in \mathbb {N}.\) Then
 (B1)
\(\lambda _k(f)\rightarrow _k 0\) for every \(f\in C(K);\)
 (B2)
\(\lambda _k(V_k)> \epsilon \) for every \(k\in \mathbb {N}\);
 (B3)
\(\lambda _k(V_l)=0\) and \(\lambda _l(V_k) \le \epsilon /3^k\) for all \(k,l\in \mathbb {N}\) with \(k<l.\)
 (B1)
 (V)Let \((x_n)\in c_0\). Define a sequence \((x'_n)\in \mathbb {R}^{\mathbb {N}}\) by the recursive formula$$\begin{aligned} x'_n:=\left[ x_n\sum _{1\le k<n}x'_k \lambda _n(\varphi _k)\right] /\lambda _n(\varphi _n) \ \text{ for } \ n\in \mathbb {N}. \end{aligned}$$
 (VI)The operator$$\begin{aligned} T: c_0 \rightarrow C_p(X),\;\; T:(x_n)\mapsto \sum _{n=1}^{\infty } x_n{\cdot }\varphi _nX, \end{aligned}$$
 (VII)Finally we prove that the quotient space \(C_p(X)/Z\) is first countable and hence metrizable. LetThe first countability of the quotient space \(C_p(X)/Z\) will follow as soon as for every neighbourhood U of zero in \(C_p(X)\) we find \(n\in \mathbb {N}\) with \(Z+U_n \subset Z+U.\) Clearly we can assume that$$\begin{aligned} U_n=\{f\in C_p(X): f(x)<1/n\;\text{ for } \text{ every }\;x\in \bigcup _{k=1}^n \mathrm {supp}(\lambda _k)\}, n\in \mathbb {N}. \end{aligned}$$for some finite subset F of X and some \(\delta >0.\)$$\begin{aligned} U=\bigcap _{x\in F}\{f\in C_p(X): f(x)<\delta \} \end{aligned}$$
Lemma 2
Let X be a Tychonoff space. Each metrizable continuous image of \(C_p(X)\) is separable.
Proof
It is wellknown [11, 2.3.18] that the Tychonoff product \(\mathbb {R}^{X}\) has countable cellularity, which means that \(\mathbb {R}^{X}\) contains no uncountable family of pairwise disjoint nonempty open sets. Then the dense subspace \(C_p(X)\) of \(\mathbb {R}^{X}\) also has countable cellularity and so does any continuous image Y of \(C_p(X)\). If Y is metrizable, then Y is separable according to Theorem 4.1.15 in [11]. \(\square \)
Lemma 3
Let X be a pseudocompact space. A closed linear subspace S of \(C_p(X)\) is separable if and only if S is Polishable.
Proof
Therefore the subspace \(C_p(M)\) of \(C_p(X)\) is closed and hence \(C_p(M)\) contains the closure S of the dense set \(\{f_n\}_{n\in \mathbb N}\) in S. Since the space \(C_p(M)\) is Polishable, so is its closed subspace S. \(\square \)
Now we are at the position to prove the main Theorem 1:
Proof of Theorem 1
 (1)
\(C_{p}(X)\) satisfies the JNP;
 (2)
\(C_p(X)\) contains a complemented subspace isomorphic to \(c_0\);
 (3)
\(C_{p}(X)\) has a quotient isomorphic to \(c_0\);
 (4)
\(C_p(X)\) admits a continuous linear map onto \(c_0.\)
We shall show that the union \(S=\bigcup _{n=1}^{\infty } S_n\) of supports \(S_n=\mathrm {supp}(\lambda _n)\) of the signmeasures \(\lambda _n\) is bounded in X in the sense that for any \(\varphi \in C_p(X)\) the image \(\varphi (S)\) is bounded in \(\mathbb {R}\), since in the opposite case we get a function \(\psi \in C_p(X)\) with \(\lambda _n (\psi )\not \rightarrow 0.\) Indeed, suppose that for some \(\varphi \in C_p(X)\) the image \(\varphi (S)\) is unbounded in \(\mathbb {R}\); without loss of generality we can assume that \(\varphi \) is nonnegative.
 (5)
\(C_p(X)\) contains a complemented infinitedimensional metrizable subspace;
 (6)
\(C_{p}(X)\) contains a complemented infinitedimensional separable subspace;
 (7)
\(C_p(X)\) has an infinitedimensional Polishable quotient.
\((7) \Rightarrow (1)\) Assume that the space \(C_p(X)\) contains a closed subspace Z of infinite codimension such that the quotient space \(E:=C_p(X)/Z\) is Polishable. Denote by \(\tau _p\) the quotient topology of \(C_p(X)/Z\) and by \(\tau _0\supset \tau _p\) a stronger separable Fréchet locally convex topology on E. Denote by \(\tau _\infty \) the topology of the quotient Banach space C(X) / Z. Here C(X) is endowed with the supnorm \(\Vert f\Vert _\infty :=\sup _{x\in X}f(x)\) (which is welldefined as X is pseudocompact).
Continuing on this way we can construct inductively a biorthogonal sequence \(((f_n, \nu _n))_{n\in \mathbb N}\) in \(C_p(X)\times C_p(X)^{*}\) such that \(\text{ lin } \{f_n: n\in \mathbb {N}\}= \text{ lin } \{g_n: n\in \mathbb {N}\}\) and \(\nu _nZ=0\), \(\nu _n(f_m)=\delta _{n,m}\) for all \(n,m \in \mathbb {N}.\) Then \(\text{ lin } \{f_n: n\in \mathbb {N}\} +Z\) is dense in \((C(X), \Vert .\Vert _{\infty })\). Let \(\mu _n=\nu _n/\Vert \nu _n\Vert \) for \( n\in \mathbb {N}.\) Then \(\Vert \mu _n\Vert =1\) and \(\mu _n(f_m)=0\) for all \(n,m \in \mathbb {N}\) with \(n\ne m.\)
4 An example of Plebanek
In this section we describe the following example suggested to the authors by Grzegorz Plebanek [26].
Example 1
 (1)
K contain no nontrivial converging sequences but contains a copy of \(\beta \mathbb {N}\);
 (2)
the function space \(C_p(K)\) has the JNP.
Fact 1
For any countable subfamily \(\mathcal C\subset \mathcal {Z}\) there is a set \(B\in \mathcal {Z}\) such that \(C\subset ^{*} B\) for all \(C\in \mathcal C\).
 (1)
\(\{n\} \in x\) for some \(n\in \mathbb {N}\); then \(x=\{A\in \mathfrak {A}: n\in A\}\) is identified with n;
 (2)
x contains no finite subsets of \(\mathbb {N}\) but \(Z\in x\) for some \(Z\in \mathcal {Z}\);
 (3)\(Z\notin x\) for every \(Z\in \mathcal {Z}\); this defines the unique$$\begin{aligned} p=\{A\in \mathfrak {A}:d(A)=1\}\in K. \end{aligned}$$
Fact 2
The space K contains no nontrivial converging sequence.
Proof
For an infinite set \(X\subset K\), we have two cases:
Case 1, \(X\cap \mathbb {N}\) is infinite. There is an infinite \(Z\subset X\cap \mathbb {N}\) having density zero. Then every subset of Z is in \(\mathfrak {A}\), which implies that \(\overline{Z}\cong \beta \mathbb {N}\) .
Case 2, \(X\cap ({K{\setminus }\mathbb {N}})\) is infinite. Let us fix a sequence of different \(x_n \in X\cap (K{\setminus }\mathbb {N})\) such that \(x_n\ne p\) for every n. Then for every n we have \(Z_n\in x_n\) for some \(Z_n\in \mathcal {Z}\). Take \(B\in \mathcal {Z}\) as in Fact 1. Then \(B\in x_n\) because \(x_n\) is a nonprincipial ultrafilter on \(\mathfrak A\) so \(A_n{\setminus }B\notin x_n\). Again, we conclude that \(\overline{\{x_n: n\in \mathbb {N}\}}\) is \(\beta \mathbb {N}\). \(\square \)
Fact 3
If \(\nu _n=\frac{1}{n}\sum _{k\le n} \delta _k\) and \(\mu _n=\frac{1}{2}(\nu _n  \delta _p)\) for \(n \in \mathbb {N}\), then \(\nu _n(f)\rightarrow _n \delta _p(f)\) and \(\mu _n(f)\rightarrow _n 0\) for every \(f\in C(K)\).
Proof
Observe \(\nu _n(A)\rightarrow _n d(A)\) for every \(A\in \mathfrak {A}\) since elements of \(\mathfrak {A}\) have asymptotic density either 0 or 1. This means that, when we treat \(\nu _n\) as measures on K then \(\nu _n(V)\) converges to \(\delta _p(V)\) for every clopen set \(V\subset K\). This implies the assertion since every continuous function on K can be uniformly approximated by simple functions built from clopens. \(\square \)
5 Proof of Theorem 2

submetrizable if X admits a continuous metric;

hemicompact if X has a countable family \(\mathcal K\) of compact sets such that each compact subset of X is contained in some compact set \(K\in \mathcal {K}\);

a kspace if a subset \(F\subset X\) is closed if and only if for every compact subset \(K\subset X\) the intersection \(F\cap K\) is closed in K.
 (1)
\(C_k(X)\) is Polishable;
 (2)
\(C_p(X)\) is Polishable;
 (3)
\(C_k(X)\) is Polish;
 (4)
X is a submetrizable hemicompact kspace.
\((2)\Rightarrow (3)\) Assume that the space \(C_p(X)\) is Polishable and fix a stronger Polish locally convex topology \(\tau \) on \(C_p(X)\). Let \(C_\tau (X)\) denote the separable Fréchet space \((C_p(X),\tau )\). By \(\tau _{k}\) denote the compact open topology of \(C_{k}(X)\). Taking into account that the space \(C_p(X)\) is a continuous image of the Polish space \(C_\tau (X)\), we conclude that \(C_p(X)\) has countable network and by [2, I.1.3], the space X has countable network and hence is Lindelöf. By the normality (and the Lindelöf property) of X, each closed bounded set in X is countably compact (and hence compact). So X is a \(\mu \)space. By Theorem 10.1.20 in [25, Theorem 10.1.20] the function space \(C_{k}(X)\) is barrelled. The continuity of the identity maps \(C_k(X)\rightarrow C_p(X)\) and \(C_\tau (X)\rightarrow C_p(X)\) implies that the identity map \(C_k(X)\rightarrow C_\tau (X)\) has closed graph. Since \(C_k(X)\) is barelled and \(C_\tau (X)\) is Fréchet, we can apply the Closed Graph Theorem 4.1.10 in [25] and conclude that the identity map \(C_k(X)\rightarrow C_\tau (X)\) is continuous.
The implication \((3)\Rightarrow (1)\) is trivial.
\((3)\Rightarrow (4)\) If the function space \(C_k(X)\) is Polish, then by Theorem 4.2 in [24], X is a hemicompact kspace. Taking into account that the space \(C_p(X)\) is a continuous image of the space \(C_k(X)\), we conclude that \(C_p(X)\) has countable network and by [2, I.1.3], the space X has countable network. By [17, 2.9], the space X is submetrizable.
The authors thank to the referee for his/her valuable comments and remarks.
Notes
References
 1.Albiac, F., Kalton, N.J.: Topics in Banach Space Theory. Springer, Berlin (2006)zbMATHGoogle Scholar
 2.Arhangel’skii, A.V.: Topological Function Spaces, Mathematics Application, vol. 78. Kluwer Academic Publishers, Dordrecht (1992)CrossRefGoogle Scholar
 3.Banakh, T., Ka̧kol, J., Śliwa, W.: Metrizable quotients of \(C_p\)spaces. Topol. Appl. 249, 95–102 (2018)CrossRefzbMATHGoogle Scholar
 4.Cembranos, P., Mendoza, J.: Banach Spaces of VectorValued Functions, Lecture Notes in Mathematics, vol. 1676. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
 5.Dales, H.D., Dashiell Jr., F.H., Lau, A.T.M., Strauss, D.: Banach Spaces of Continuous Functions as Dual Spaces. Springer, Berlin (2016)CrossRefzbMATHGoogle Scholar
 6.Diestel, J.: Sequences and Series in Banach spaces. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
 7.Dow, A.: Efimov spaces and the splitting number. Topol. Proc. 29, 105–113 (2005)MathSciNetzbMATHGoogle Scholar
 8.Dow, A.: Compact sets without converging sequences in the random real model. Acta Math. Univ. Comenianae 76, 161–171 (2007)MathSciNetzbMATHGoogle Scholar
 9.Dow, A., Shelah, S.: An Efimov space from Martin’s Axiom. Houston J. Math. 39, 1423–1435 (2013)MathSciNetzbMATHGoogle Scholar
 10.Efimov, B.: Subspaces of dyadic bicompacta. Doklady Akademiia Nauk USSR 185, 987–990 (1969). ((Russian); English transl., Soviet Mathematics. Doklady 10, 453–456 (1969))MathSciNetzbMATHGoogle Scholar
 11.Engelking, R.: General Topology. Heldermann, Berlin (1989)zbMATHGoogle Scholar
 12.Fedorchuk, V.V.: A bicompactum whose infinite closed subsets are all ndimensional. Math. USSR Sbornik 25, 37–57 (1976)CrossRefzbMATHGoogle Scholar
 13.Fedorchuk, V.V.: Completely closed mappings, and the consistency of certain general topology theorems with the axioms of set theory. Math. USSR Sbornik 28, 3–33 (1976)Google Scholar
 14.Ferrando, J.C., Ka̧kol, J., LopezPellicer, M., Śliwa, W.: On the separable quotient problem for Banach spaces. Funct. Approx. 59, 153–173 (2018)MathSciNetCrossRefGoogle Scholar
 15.Ferrando, J.C., Ka̧kol, J., Saxon, S.A.: The dual of of the locally convex space \(C_{p}(X)\). Funct. Approx. 50, 1–11 (2014)Google Scholar
 16.Geschke, S.: The coinitialities of Efimov spaces, set theory and its applications, Babinkostova et al., Editors. Contemp. Math. 533, 259–265 (2011)CrossRefzbMATHGoogle Scholar
 17.Gruenhage, G.: Generalized metric spaces. Handbook of SetTheoretic Topology, pp. 423–501. NorthHolland, Amsterdam (1984)CrossRefGoogle Scholar
 18.Hart, P.: Efimov’s problem. In: Pearl, E. (ed.) Problems in Topology II, pp. 171–177. Elsevier, Oxford (2007)Google Scholar
 19.Johnson, W.B., Rosenthal, H.P.: On weak*basic sequences and their applications to the study of Banach spaces. Stud. Math. 43, 166–168 (1975)Google Scholar
 20.Kechris, A.S.: Classical descriptive set theory. Grad. Texts Math. 156 (1995)Google Scholar
 21.Ka̧kol, J., Saxon, S.A.: Separable quotients in \(C_{c}(X)\), \(C_{p}(X)\) and their duals. Proc. Am. Math. Soc. 145, 3829–3841 (2017)CrossRefzbMATHGoogle Scholar
 22.Ka̧kol, J., Śliwa, W.: Efimov spaces and the separable quotient problem for spaces \(C_p (K)\). J. Math. Anal. App. 457, 104–113 (2018)CrossRefzbMATHGoogle Scholar
 23.Lacey, E.: Separable quotients of Banach spaces. An. Acad. Brasil. Ciènc. 44, 185–189 (1972)MathSciNetzbMATHGoogle Scholar
 24.McCoy, R.A., Ntantu, I.: Completeness properties of function spaces. Topol. Appl. 22(2), 191–206 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
 25.Pérez Carreras, P., Bonet, J.: Barrelled Locally Convex Spaces, NorthHolland Mathematics Studies, vol. 131. NorthHolland, Amsterdam (1987)zbMATHGoogle Scholar
 26.Plebanek, G.: On Some Extremally Disconnected Compact Space. Private Commun. (2018)Google Scholar
 27.Rosenthal, H.P.: On quasicomplemented subspaces of Banach spaces, with an appendix on compactness of operators from \(L_{p}(\mu )\) to \(L_{r}(\nu )\). J. Funct. Anal. 4, 176–214 (1969)CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.