Classifying Topologies through $\mathfrak{G}$-Bases

Abstract

We classify several topological properties of a Tychonoff space X by means of certain locally convex topologies $\mathcal{T}$ with a $\mathfrak{G}$-base located between the pointwise topology ${\tau }_p$ and the bounded-open topology ${\tau }_b$ of the real-valued continuous function space $C\left(X\right)$.

1. Introduction

Unless otherwise stated, X stands for an infinite Tychonoff space. We denote by $C_p\left(X\right)$ the linear space $C\left(X\right)$ of real-valued continuous functions on X equipped with the pointwise topology ${\tau }_p$. The topological dual of $C_p\left(X\right)$ is denoted by $L\left(X\right)$, or by $L_p\left(X\right)$ when provided with the weak* topology. We represent by $C_k\left(X\right)$ the space $C\left(X\right)$ equipped with the compact-open topology ${\tau }_k$. A family $\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ of subsets of a set X is called a resolution for X if it covers X and verifies that $A_{\alpha }\subseteq A_{\beta }$ for $\alpha \le \beta$. Let us recall that a Hausdorff topological space X is called a $\mu$-space if the closure of each functionally bounded set in X is compact. Every realcompact space is a $\mu$-space, but the converse is not true. If E is a locally convex space, the bidual $E^{″}$ of E is the topological dual of the strong dual $E_{\beta }^{\prime }$ of E. By the strong dual $E_{\beta }^{\prime }$ of E, we mean the dual $E^{\prime }$ of E equipped with the strong topology $\beta \left(E^{\prime },E\right)$ of the uniform convergence on the bounded sets in E. We denote by ${\tau }_w$ and ${\tau }_b$ the weak locally convex topology of $C_k\left(X\right)$ and the bounded-open topology of $C\left(X\right)$, respectively. If A is a nonempty set in a real linear space L, we represent by $\mathrm{abx}\left(A\right)$ the (real) absolutely convex hull of A. The linear subspace of $C\left(X\right)$ consisting of those bounded functions is denoted by $C^b\left(X\right)$.

A base $\{U_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ of (absolutely convex) neighborhoods of the origin in a locally convex space E such that $U_{\beta }\subseteq U_{\alpha }$ if $\alpha \le \beta$ is called a $\mathfrak{G}$-base. Let us mention that the notion of a $\mathfrak{G}$-base, originally introduced in the realm of locally convex spaces [1], has been extended to topological groups and general topological spaces, sometimes under the name of ${\omega }^{\omega }$-base, by some authors (see [2,3]). However, in this paper we keep the original name. Trivially, if E is a metrizable locally convex space with a decreasing base $\left\{V_n:n\in \mathbb{N}\right\}$ of locally convex neighborhoods of the origin, the family $\{U_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$, where $U_{\alpha }=V_{\alpha \left(1\right)}$ for all $\alpha \in {\mathbb{N}}^{\mathbb{N}}$ is a $\mathfrak{G}$-base for E. The converse is not true, i.e., a locally convex space with a $\mathfrak{G}$-base need not be metrizable (see p. 107, [4]).

A locally convex space E with a $\mathfrak{G}$-base has metrizable compact sets, since, in this case, the weak dual $\left(E^{\prime },\sigma \left(E^{\prime },E\right)\right)$ is quasi-Suslin [5], hence trans-separable [6] (properties not defined in this paper can be found in [7,8,9,10]). Therefore, if there is a locally convex topology $\mathcal{T}$ on $C\left(X\right)$ with a $\mathfrak{G}$-base such that ${\tau }_p\le \mathcal{T}\le {\tau }_b$, every completely regular topology $\tau$ on $C\left(X\right)$ such that $\mathcal{T}\le \tau$ is angelic and has metrizable compact sets. Research on $\mathfrak{G}$-bases and their generalizations remain active since [1]. For recent results on this topic see [3,11,12,13,14]. Research on the existence of locally convex topologies on $C\left(X\right)$ stronger than the pointwise topology has been studied in [15,16]. In this paper, we enlarge the classification of topological properties on X provided in (Theorem 3.1, [15]).

2. A Preliminary Result

A resolution $\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ for X is functionally bounded if it consists of functionally bounded sets $A_{\alpha }$ in X. A space X is said to be strictly angelic [17] if it is angelic and all separable compact subsets of X are first countable. A classification of topologies on X by locally convex topologies on $C_p\left(X\right)$ is provided in [15] (see also (Theorem 98, [4])).

Theorem 1 

(Ferrando–Gabriyelyan–Kąkol (Theorem 3.1, [15])). If X is a Tychonoff space, the following properties hold

1. 

There exists a metrizable locally convex topology $\mathcal{T}$ on $C\left(X\right)$ such that ${\tau }_p\le \mathcal{T}\le {\tau }_k$ if and only if X is a σ-compact space.

2. 

There exists a metrizable locally convex topology $\mathcal{T}$ on $C\left(X\right)$ such that ${\tau }_k\le \mathcal{T}\le {\tau }_b$ if and only if there is an increasing sequence $\{A_n:n\in \mathbb{N}\}$ of functionally bounded subsets of X swallowing the compact sets of X.

3. 

There exists a metrizable locally convex topology $\mathcal{T}$ on $C\left(X\right)$ such that ${\tau }_p\le \mathcal{T}\le {\tau }_b$ if and only if there is an increasing sequence $\{A_n:n\in \mathbb{N}\}$ of functionally bounded subsets of X covering X, or equivalently, if and only if $\upsilon X$ is σ-compact.

4. 

There is a metrizable locally convex topology $\mathcal{T}$ on $C\left(X\right)$ such that ${\tau }_p\le \mathcal{T}\le {\tau }_w$ if and only if X is countable.

5. 

There is a locally convex topology $\mathcal{T}$ on $C\left(X\right)$ with a $\mathfrak{G}$-base such that ${\tau }_p\le \mathcal{T}\le {\tau }_k$ if and only if X has a compact resolution.

6. 

There is a locally convex topology $\mathcal{T}$ on $C\left(X\right)$ with a $\mathfrak{G}$-base such that ${\tau }_k\le \mathcal{T}\le {\tau }_b$ if and only if X has a functionally bounded resolution swallowing the compact sets.

7. 

There exists a locally convex topology $\mathcal{T}$ on $C\left(X\right)$ with a $\mathfrak{G}$-base such that ${\tau }_p\le \mathcal{T}\le {\tau }_b$ if and only if X has a functionally bounded resolution. Equivalently, if and only if $\upsilon X$ is K-analytic. In this case $\left(C\left(X\right),{\tau }_b\right)$ is strictly angelic.

8. 

There is a locally convex topology $\mathcal{T}$ on $C\left(X\right)$ with a $\mathfrak{G}$-base such that ${\tau }_p\le \mathcal{T}\le {\tau }_w$ if and only if X is countable.

3. A More Complete Classification

To enlarge the previous classification, we need the following result.

Lemma 1. 

Let X be completely regular. If Q is a metrizable and compact subspace of X, there exists a continuous linear extender map $\phi :C_k\left(Q\right)\to C_k\left(X\right)$, i.e., such that ${\left.\phi \left(f\right)\right|}_Q=f$ for every $f\in C\left(Q\right)$.

Proof. 

Since Q is (homeomorphic to) a metrizable compact subspace of the Stone–Čech compactification $\beta X$ of X, according to [18] (see also (Proposition 4.1, [19])) there exists a continuous linear map $\psi :C_p\left(Q\right)\to C_p\left(\beta X\right)$ such that ${\left.\psi \left(f\right)\right|}_Q=f$ which embeds $C_p\left(Q\right)$ in $C_p\left(\beta X\right)$ (as a closed subspace). If $\varphi :C_p\left(\beta X\right)\to C_p^b\left(X\right)$ is the restriction map $\varphi \left(g\right)={\left.g\right|}_X$, it turns out that $\phi :=\varphi \circ \psi$ is a continuous linear map from $C_p\left(Q\right)$ into $C_p^b\left(X\right)$ such that ${\left.\phi \left(f\right)\right|}_Q={\left.\varphi \left(\psi \left(f\right)\right)\right|}_Q={\left.\psi \left(f\right)\right|}_Q=f$ for every $f\in C\left(Q\right)$.

Hence, $\phi :C\left(Q\right)\to C^b\left(X\right)$ has a closed graph when both spaces are regarded as Banach spaces. This implies in particular that $\phi :C_k\left(Q\right)\to C_k^b\left(X\right)$ is continuous, since the supremum norm topology on $C^b\left(X\right)$ is stronger than the relative compact-open topology of $C_k\left(X\right)$. Thus, $\phi :C_k\left(Q\right)\to C_k\left(X\right)$ is a continuous linear extender map, as stated. □

Theorem 2. 

If X is a Tychonoff space, the following properties hold

1. 

The compact-open topology $\mathcal{T}={\tau }_k$ on $C\left(X\right)$ is metrizable if and only if X is a hemicompact space (Arens’ theorem (Theorem 7, [20]), see also (Theorem 2.5, [21])).

2. 

The bounded-open topology $\mathcal{T}={\tau }_b$ on $C\left(X\right)$ is metrizable if and only if there is an increasing sequence $\{A_n:n\in \mathbb{N}\}$ of functionally bounded subsets of X swallowing the functionally bounded sets of X.

3. 

The weak topology $\mathcal{T}={\tau }_w$ on $C\left(X\right)$ is metrizable if and only if X is countable and compact sets in X are finite.

4. 

The pointwise topology $\mathcal{T}={\tau }_p$ on $C\left(X\right)$ is metrizable if and only X is countable.

5. 

The compact-open topology $\mathcal{T}={\tau }_k$ on $C\left(X\right)$ has a $\mathfrak{G}$-base if and only if X has a compact resolution that swallows the compact sets (Theorem 2, [22]).

6. 

The bounded-open topology $\mathcal{T}={\tau }_b$ on $C\left(X\right)$ has a $\mathfrak{G}$-base if and only if X has a functionally bounded resolution that swallows the functionally bounded sets.

7. 

The weak topology $\mathcal{T}={\tau }_w$ on $C\left(X\right)$ has a $\mathfrak{G}$-base if and only if X is countable and compact sets in X are finite.

8. 

The pointwise topology $\mathcal{T}={\tau }_p$ on $C\left(X\right)$ has a $\mathfrak{G}$-base if and only X is countable (Corollary 15.2, [7]).

Proof. 

The proof of statements $\left(2\right)$ and $\left(6\right)$ is similar (for $\left(6\right)$ see (Theorem 12, [23])).

Let us prove statement $\left(3\right)$. If $C_k\left(X\right)$ is weakly metrizable, the weakly bounded sets are metrizable, so X is countable by (Theorem 2.3, [21]). This fact also follows from statement $\left(4\right)$ of Theorem 1. We claim that, in addition, the compact sets of X are finite. Otherwise, there exists an infinite compact set Q in X. However, since X is countable, Q is metrizable. By Lemma 1 there is a continuous linear extender $\phi$ from $C_k\left(Q\right)$ into $C_k\left(X\right)$, i.e., such that ${\left.\phi \left(f\right)\right|}_Q=f$ for every $f\in C\left(Q\right)$. If $\phi \left(f_d\right)\to g$ in $C_k\left(X\right)$, given $ϵ>0$ there is $h\in D$ with ${sup}_{x\in Q}\left|\phi \left(f_d\right)\left(x\right)-g\left(x\right)\right|<ϵ$ for every $d\ge h$, so ${sup}_{x\in Q}\left|f_d\left(x\right)-g\left(x\right)\right|<ϵ$ for $d\ge h$. Thus, $f_d\to f:={\left.g\right|}_Q$ in $C_k\left(Q\right)$, and hence $\phi \left(f_d\right)\to \phi \left(f\right)$ in $C_k\left(X\right)$, i.e., $g=\phi \left(f\right)$, which means that $\phi$ embeds the Banach space $C_k\left(Q\right)$ in $C_k\left(X\right)$ as a closed subspace. Since the weak topology is inherited by linear subspaces, the space $C_w\left(Q\right)$ is linearly homeomorphic to a subspace of $C_w\left(X\right)$. In other words, the Banach space $C_k\left(Q\right)$ is weakly metrizable. As the weak topology of a Banach space is metrizable if and only if it is finite-dimensional, it turns out that Q must be finite, a contradiction. Conversely, if X is both countable and has finite compact sets, the former statement guarantees that $C_p\left(X\right)$ is metrizable whereas the latter implies that $C_p\left(X\right)=C_k\left(X\right)$ coincides with $C_w\left(X\right)$. Hence, the weak topology ${\tau }_w$ is metrizable.

The proof of statement $\left(7\right)$ is similar to that of statement $\left(3\right)$. The only difference is that the weak topology of the Banach space $C_k\left(Q\right)$ now carries a $\mathfrak{G}$-base. However, if a locally convex space in its weak topology has a $\mathfrak{G}$-base, it is countable-dimensional (Proposition 11.2, [7]). Thus, $C\left(Q\right)$ must be countable-dimensional, so finite-dimensional by the Baire category theorem. This ensures that the compact set Q must be finite. □

Remark 1. 

In (Theorem 2.3, [21]), it is shown that if X is countable, the bounded sets of $C_k\left(X\right)$ are weakly metrizable. According to statement $\left(3\right)$ of the previous theorem, if X contains an infinite compact set, then $C_k\left(X\right)$ cannot be weakly metrizable. Thus, if X is countable and contains an infinite compact set, then $C_w\left(X\right)$ is not metrizable but has metrizable bounded sets.

Example 1. 

As $\mathbb{Q}$ is a countable space with infinite compact sets which is not hemicompact, neither ${\tau }_k$ nor ${\tau }_w$ are metrizable, the former statement by Arens’ theorem and the latter by the previous remark, but ${\tau }_p$ is metrizable. Of course, ${\tau }_w$ has no $\mathfrak{G}$-base. In fact, since $\mathbb{Q}$ is not a Polish space, Christensen’s theorem (Theorem 94, [4]) prevents $\mathbb{Q}$ from having a compact resolution that swallows the compact sets. This also implies that $C_k\left(\mathbb{Q}\right)$ has no $\mathfrak{G}$-base, by statement $\left(5\right)$ of Theorem 2.

Example 2. 

Let $\mathbb{N}$ be equipped with the discrete topology and choose $p\in \beta \mathbb{N}\backslash \mathbb{N}$. Then, $X:=\mathbb{N}\cup \left\{p\right\}$ provided with the relative topology of $\beta \mathbb{N}$ is not discrete but has finite compact sets. Thus, the weak topology ${\tau }_w$ on $C\left(X\right)$ is metrizable. In fact, clearly, ${\tau }_p={\tau }_k$, although $C_p\left(X\right)\ne {\mathbb{R}}^X$.

Example 3. 

$\mathbb{R}$ is an uncountable hemicompact space, hence ${\tau }_k$ is metrizable, but ${\tau }_w$ and ${\tau }_p$ are not.

Example 4. 

If K is a countable infinite compact set, obviously both ${\tau }_p$ and ${\tau }_k$ are metrizable, but ${\tau }_w$ is not.

Example 5. 

The Sorgenfrey line $\mathbb{S}$ is a Lindelöf space which is not σ-compact, so ${\tau }_k={\tau }_b$ and by statement $\left(1\right)$ of Theorem 1 there is no metrizable, locally convex topology $\mathcal{T}$ on $C\left(\mathbb{S}\right)$ such that ${\tau }_p\le \mathcal{T}\le {\tau }_k$.

Example 6. 

The space $C_k\left({\mathbb{N}}^{\mathbb{N}}\right)$ has a $\mathfrak{G}$-base by statement $\left(5\right)$ of Theorem 2, but there is no metrizable topology $\mathcal{T}$ such that ${\tau }_p\le \mathcal{T}\le {\tau }_k$, since ${\mathbb{N}}^{\mathbb{N}}$ is not σ-compact.

4. The Interval ${\mathbf{\tau }}_{\mathbf{w}}\le \mathcal{T}\le {\mathbf{\tau }}_{\mathbf{k}}$

In this section, we deal with the interval ${\tau }_w\le \mathcal{T}\le {\tau }_k$. Before stating our main result for this case, we need to establish two auxiliary results concerning the weak* dual $L_p\left(X\right)$ of the space $C_p\left(X\right)$. We regard (the canonical homeomorphic copy $\delta \left(X\right)$ of) X in $L_p\left(X\right)$ as a Hamel basis of $L\left(X\right)$, and denote by ${\delta }_x$ the image of $x\in X$ in $\delta \left(X\right)$. If $f\in C\left(X\right)$ and $u\in L\left(X\right)$, we write $〈f,u〉$ to represent the action of the linear functional u on f, in particular $〈f,{\delta }_x〉=f\left(x\right)$.

Lemma 2. 

Let E denote the dual of $C_k\left(X\right)$. If X is a μ-space, then E coincides with the bidual of $L_p\left(X\right)$.

Proof. 

If X is a $\mu$-space, then $C_k\left(X\right)$ is the strong dual of $L_p\left(X\right)$ (see (Lemma 2.2, [24])). Therefore, $C_k\left(X\right)=L{\left(X\right)}_{\beta }^{\prime }$ and hence, E coincides with the bidual $L_p{\left(X\right)}^{″}$ of $L_p\left(X\right)$. □

Lemma 3. 

Let X be a μ-space. If A is a bounded set in $L_p\left(X\right)$, there are a compact set $K_A$ in X and a real number ${ϵ}_A$ with $0<{ϵ}_A<1$ such that $A\subseteq 2{ϵ}_A^{-1}\mathrm{abx}\left(K_A\right)$.

Proof. 

Since X is a $\mu$-space, if A is a bounded set in $L_p\left(X\right)$, then $A^0$ is a neighborhood of the origin in $C_k\left(X\right)$, so there is $0<{ϵ}_A<1$ and a compact set $K_A$ in X such that

$$U_A:=\{f\in C\left(X\right):{sup}_{x\in K_A}\left|f\left(x\right)\right|\le {ϵ}_A\}\subseteq A^0$$

which implies that $A\subseteq A^{00}\subseteq U_A^0$, where the bipolar of A is taken in $L\left(X\right)$ as well as the polar of $U_A$. The fact that $U_A={ϵ}_A{K}_A^0$ yields ${ϵ}_{A}A\subseteq K_A^{00}$ so that ${ϵ}_{A}A\subseteq {\overline{\mathrm{abx}\left(K_A\right)}}^{L_p\left(X\right)}$. We claim that ${\overline{\mathrm{abx}\left(K_A\right)}}^{L_p\left(X\right)}\subseteq 2\mathrm{abx}\left(K_A\right)$.

Assume that $u\in {\overline{\mathrm{abx}\left(K_A\right)}}^{L_p\left(X\right)}=\mathrm{abx}{\left(K_A\right)}^{00}$ with $u\ne \mathbf{0}$. Since X is a Hamel basis of $L\left(X\right)$, we have that $u={\sum }_{i=1}^m{a}_i{\delta }_{x_i}$, with $a_i\ne 0$ for $1\le i\le m$. First, we show that the support of u is contained in $K_A$, so that $x_i\in K_A$ for $1\le i\le m$. Indeed if $x_k\notin K_A$ for some $k\in \left\{1,\dots ,m\right\}$, there is $f\in C\left(X\right)$ such that $f\left(x_k\right)=2a_k^{-1}$ and $f\left(z\right)=0$ for every $z\in K_A\cup \left\{x_i:1\le i\le m,i\ne k\right\}$. Note that $f\in \mathrm{abx}{\left(K_A\right)}^0$ since if $v={\sum }_{i=1}^n{b}_i{\delta }_{y_i}\in \mathrm{abx}\left(K_A\right)$ with ${\sum }_{i=1}^n\left|b_i\right|\le 1$ and $y_i\in K_A$ for $1\le i\le n$, then $〈f,v〉={\sum }_{i=1}^n{b}_{i}f\left(y_i\right)=0$. As $〈f,u〉=2$, this contradicts the fact that $u\in \mathrm{abx}{\left(K_A\right)}^{00}$. Thus, $x_i\in K_A$ for $1\le i\le m$.

Now, we show that ${\sum }_{i=1}^m\left|a_i\right|\le 2$, so that $u\in 2\mathrm{abx}\left(K_A\right)$. Since the support of u is finite and is contained in $K_A$, there are pairwise disjoint open sets $\left\{U_1,\dots ,U_m\right\}$ in $K_A$ with $x_i\in U_i$, so there are compactly supported continuous functions $\left\{f_1,\dots ,f_m\right\}$ such that $\left\{x_i\right\}\prec f_i\prec U_i$ for $1\le i\le m$. This means that $f_i\in C\left(X\right)$, $f_i$ has compact support, $0\le f_i\le 1$, $\mathrm{supp}{f}_i\subseteq U_i$ and $f\left(x_i\right)=1$ for $1\le i\le m$. For $1\le i\le m$, set $h_i=\mathrm{sgn}\left(a_i\right)f_i$, where $\mathrm{sgn}\left(a_i\right)\in \left\{-1,1\right\}$ denotes the sign of the real number $a_i$, and define $h:={\sum }_{i=1}^m{h}_i$. Clearly $h\in C\left(X\right)$, $h\left(x_i\right)=\mathrm{sgn}\left(a_i\right)$, and $\left|h\left(x\right)\right|\le 1$ for every $x\in X$, because of the supports of the functions $f_i$ are pairwise disjoint.

Since $u\in {\overline{\mathrm{abx}\left(K_A\right)}}^{L_p\left(X\right)}$ and $h\in C\left(X\right)$ there exists $v={\sum }_{i=1}^k{c}_i{\delta }_{z_i}\in \mathrm{abx}\left(K_A\right)$ with ${\sum }_{i=1}^k\left|c_i\right|\le 1$ and $z_i\in K_A$ for $1\le i\le k$ such that $\left|〈h,u-v〉\right|\le 1$. Consequently,

$$\left|\sum _{i=1}^m{a}_{i}h\left(x_i\right)-\sum _{j=1}^k{c}_{j}h\left(z_j\right)\right|\le 1,$$

and due to the fact that $v\in \mathrm{abx}\left(K_A\right)$ and $a_{i}h\left(x_i\right)=\left|a_i\right|$ it follows that

$$\sum _{i=1}^m\left|a_i\right|=\left|\sum _{i=1}^m{a}_{i}h\left(x_i\right)\right|\le 1+\left|\sum _{j=1}^k{c}_{j}h\left(z_j\right)\right|\le 1+\sum _{j=1}^k\left|c_j\right|\left|h\left(z_j\right)\right|\le 2$$

as required. □

If X is a $\mu$-space and $\mathsf{Bound}\left(X\right)$ denotes the family of all bounded sets in $L_p\left(X\right)$, as a consequence of Lemma 2, the space E coincides with (see (23.2.(1), [8]))

$$E=\bigcup \{\overline{A}:A\in \mathsf{Bound}\left(X\right)\}$$

where the closure is in E under the weak* topology $\sigma \left(E,C\left(X\right)\right)$ of E. Hence, if $\mu \in E$ there is a bounded set A in $L_p\left(X\right)$ with $\mu \in \overline{A}$. However according to Lemma 3, given $A\in \mathsf{Bound}\left(X\right)$, there is a compact $S_A$ in X together with some $n_A\in \mathbb{N}$ with $A\subseteq n_A\mathrm{abx}\left(S_A\right)$. Thus, if $\mathcal{B}\left(X\right)$ designates the family of all functionally bounded sets in X, then

$$E=\bigcup \{n_A\overline{\mathrm{abx}\left(S_A\right)}:A\in \mathsf{Bound}\left(X\right)\}=\bigcup \{n\overline{\mathrm{abx}\left(S\right)}:S\in \mathcal{B}\left(X\right),n\in \mathbb{N}\}$$

where the closures are in $\sigma \left(E,C\left(X\right)\right)$. Therefore, the following property holds.

Remark 2. 

If X is a μ-space and $\mathcal{K}\left(X\right)$ stands for the family of all compact sets in X, then

$$E=\bigcup \left\{n{S}^{00}:S\in \mathcal{K}\left(X\right),n\in \mathbb{N}\right\}$$

where E is the dual of $C_k\left(X\right)$.

Proof. 

Because of the bipolar theorem, one has $\overline{\mathrm{abx}\left(S\right)}=S^{00}$. □

This motivates the following definition.

Definition 1. 

Let E be the dual of $C_k\left(X\right)$. We say that a countable family $\mathcal{F}=\left\{S_n:n\in \mathbb{N}\right\}$ of compact sets in X is a complete sequence if there exists a sequence $\left\{k_n:n\in \mathbb{N}\right\}$ of positive integers such that

$$E={\bigcup }_{n\in \mathbb{N}}{k}_n{S}_n^{00}.$$

If X is a metrizable hemicompact space and $\mathcal{F}=\left\{S_n:n\in \mathbb{N}\right\}$ is a family of compact sets that swallows the compact sets in X, then $\mathcal{F}$ is a complete sequence. If $\left\{V_j:j\in \mathbb{N}\right\}$ is a base of neighborhoods of the origin in $C_k\left(X\right)$, there are by Lemma 3 a compact set S in X and $k_j\in \mathbb{N}$ such that $V_j^0\subseteq k_j\mathrm{abx}\left(S\right)$. If $S\subseteq S_{n_j}$ then $V_j^0\subseteq k_j{S}_{n_j}^{00}$ for every $j\in \mathbb{N}$. If we define $k_m=k_j$ if $m=n_j$ for some $j\in \mathbb{N}$ and $k_m=1$ otherwise, then

$$E={\bigcup }_{j\in \mathbb{N}}{V}_j^0={\bigcup }_{m\in \mathbb{N}}{k}_m{S}_m^{00}.$$

If $〈F,E〉$ is a dual pair, a locally convex topology $\mathcal{T}$ for F is called a polar topology if $\mathcal{T}$ is the topology of the uniform convergence on the sets of a family $\mathcal{A}$ of $\sigma \left(E,F\right)$-bounded sets in E. Obviously, both the weak topology $\sigma \left(F,E\right)$ and the strong topology $\beta \left(F,E\right)$ are polar topologies for F. We are ready to establish our last classification result, which reads as follows.

Theorem 3. 

Let X be a Tychonoff space. There is a metrizable polar topology $\mathcal{T}$ on $C\left(X\right)$ such that ${\tau }_w\le \mathcal{T}\le {\tau }_k$ if and only if X contains a complete sequence.

Proof. 

Let us denote by E the dual of the locally convex space $C_k\left(X\right)$. Assume that $\mathcal{T}$ is a metrizable polar topology on $C\left(X\right)$ with a decreasing base $\{U_n:n\in \mathbb{N}\}$ of neighborhoods of the origin enjoying that ${\tau }_w\le \mathcal{T}\le {\tau }_k$. By the first statement of Theorem 1, we know that X is a $\sigma$-compact space, so a $\mu$-space.

As $\mathcal{T}$ is a polar topology, there exists a family $\mathcal{M}$ of bounded sets in $L_p\left(X\right)$ such that for each $n\in \mathbb{N}$, there are some $M_n\in \mathcal{M}$ with $U_n=M_n^0$. According to Lemma 3, there is a compact set $S_n$ in X and some $k_n\in \mathbb{N}$ such that $M_n\subseteq k_n\mathrm{abx}\left(S_n\right)$, so that $S_n^0\subseteq k_n{U}_n$. Set $\mathcal{F}:=\left\{S_n:n\in \mathbb{N}\right\}$. Since $U_n^0\subseteq k_n{S}_n^{00}$ for each $n\in \mathbb{N}$ and ${\tau }_w\le \mathcal{T}$, we get

$${\bigcup }_{n\in \mathbb{N}}{k}_n{S}_n^{00}={\bigcup }_{n\in \mathbb{N}}{U}_n^0=E.$$

Thus, the sequence $\mathcal{F}$ is complete.

Assume conversely that there is a complete sequence $\mathcal{F}=\{S_n:n\in \mathbb{N}\}$ in X. We may suppose that $S_n\subseteq S_{n+1}$ as well as that $k_n\le k_{n+1}$. If not, replace $\mathcal{F}$ by the sequence ${\mathcal{F}}_1:=\left\{S_n^{\prime }:n\in \mathbb{N}\right\}$ with $S_n^{\prime }={\bigcup }_{i=1}^n{S}_i$ and the sequence $\left\{k_n:n\in \mathbb{N}\right\}$ by $\left\{k_n^{\prime }:n\in \mathbb{N}\right\}$ with $k_n^{\prime }={max}_{1\le i\le n}{k}_i$. Since $S_n^{00}\subseteq {S^{\prime }}_n^{00}$ for every $n\in \mathbb{N}$, if $\mathcal{F}$ is complete so is ${\mathcal{F}}_1$. Then, for each $n\in \mathbb{N}$, define $V_n=k_n^{-1}{S}_n^0$. Clearly, $V_{n+1}\subseteq V_n$ and $\{V_n:n\in \mathbb{N}\}$ is a base of neighborhoods of the origin of a metrizable locally convex topology $\mathcal{T}$ for $C\left(X\right)$. Observe that $\mathcal{T}$ is stronger than ${\tau }_w$, since

$${\bigcup }_{n\in \mathbb{N}}{V}_n^0={\bigcup }_{n\in \mathbb{N}}{k}_n{S}_n^{00}=E.$$

On the other hand, as $\mathcal{F}\subseteq \mathcal{K}\left(X\right)$, the family of all compact sets in X, we see that $\mathcal{T}$ is weaker than ${\tau }_k$. □

5. Conclusions

We enlarged the classification of some topological properties on X provided in (Theorem 3.1, [15]) by using specific locally convex topologies $\mathcal{T}$ with a $\mathfrak{G}$-base lying between the pointwise topology ${\tau }_p$ and the bounded-open topology ${\tau }_b$ of the real-valued continuous function space $C\left(X\right)$. Our main results are

Theorem 4. 

Let X be a Tychonoff space.

1. 

The weak topology $\mathcal{T}={\tau }_w$ on $C\left(X\right)$ is metrizable if and only if it has a $\mathfrak{G}$-base, and if and only if X is countable and compact sets in X are finite.

2. 

There is a metrizable polar topology $\mathcal{T}$ on $C\left(X\right)$ such that ${\tau }_w\le \mathcal{T}\le {\tau }_k$ if and only if X contains a complete sequence.

In the second statement, we say that a countable family $\mathcal{F}=\left\{S_n:n\in \mathbb{N}\right\}$ of compact sets in X is a complete sequence if there exists a sequence $\left\{k_n:n\in \mathbb{N}\right\}$ of positive integers such that $\left\{k_n{S}_n^{00}:n\in \mathbb{N}\right\}$ covers the dual of $C_k\left(X\right)$.