Josefson–Nissenzweig property for \(C_{p}\)spaces
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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
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