Compact Resolutions and Analyticity

Abstract

We consider the large class $\mathfrak{G}$ of locally convex spaces that includes, among others, the classes of $\left(DF\right)$-spaces and $\left(LF\right)$-spaces. For a space E in class $\mathfrak{G}$ we have characterized that a subspace Y of $(E,\sigma (E,E^{\prime }))$, endowed with the induced topology, is analytic if and only if Y has a $\sigma (E,E^{\prime })$-compact resolution and is contained in a $\sigma (E,E^{\prime })$-separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class $\mathfrak{G}$. The mentioned characterization follows from the following analogous result: The space $C\left(X\right)$ of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology $\xi$ stronger or equal than the pointwise topology ${\tau }_p$ of $C\left(X\right)$ is analytic iff $\left(C\right(X),\xi )$ is separable and is covered by a compact resolution.

1. Introduction

A family $\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ of sets covering a set X is called a resolution of X if $A_{\alpha }\subset A_{\beta }$ whenever $\alpha \le \beta$, $\alpha ,\beta \in {\mathbb{N}}^{\mathbb{N}}$. A locally convex topological vector space E belongs to class $\mathfrak{G}$ if there is a resolution $\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ in $(E^{\prime },\sigma (E^{\prime },E))$ such that each sequence in any $A_{\alpha }$ is equicontinuous [1], and the resolution $\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ is called a $\mathfrak{G}$-representation of $E^{\prime }$.

The class $\mathfrak{G}$ is stable by taking subspaces, Hausdorff quotients, countable direct sums, and products. It contains “almost all” important classes of locally convex spaces, including $\left(LF\right)$-spaces and $\left(DF\right)$-spaces, hence it is indeed a very large class. We recall that this class $\mathfrak{G}$ of locally convex space was introduced in [1] motivated by particular results for $\left(LF\right)$-spaces and $\left(DF\right)$-spaces and common properties of the topological dual of each space of these two classes.

An interesting result from [1] states that a compact set K is Talagrand compact if and only if it is homeomorphic to a subset of a locally convex space in class $\mathfrak{G}$. Therefore, dealing with Talagrand compact sets, one may ask when (weakly) compact sets in a locally convex space in class $\mathfrak{G}$ are (weakly) metrizable. Both questions were answered in [1,2], respectively, see also [3] (and references there). Additionally, in the theory of locally convex spaces working with compact sets of a locally convex space E raise the questions about metrizability and weakly angelicity of compact subsets of E. In [1] and references therein, a list of positive results concerning both questions is provided, with $\left(LF\right)$-spaces and $\left(DF\right)$-spaces included in the list. For the spaces in class $\mathfrak{G}$, both above-mentioned problems have positive answers.

Nevertheless, as was proved in [4], the space $C_p\left(X\right)$ of continuous real-valued maps on a completely regular Hausdorff space X, endowed with the pointwise topology belongs to class $\mathfrak{G}$ if and only if $C_p\left(X\right)$ is metrizable.

All topological spaces are assumed to be completely regular. A topological space X is web-bounding [5] (Note 3) if there is a family $\{A_{\alpha }:\alpha \in \mathsf{\Omega }\}$ of subsets of X for some non-empty $\mathsf{\Omega }\subset {\mathbb{N}}^{\mathbb{N}}$ whose union $X_0$ is dense in X and such that if $\alpha =\left(n_k\right)\in \mathsf{\Omega }$ and $x_k\in C_{n_1,n_2,\dots ,n_k}:=\bigcup \{A_{\beta }:\beta =\left(m_k\right)\in \mathsf{\Omega },m_j=n_j,j=1,\dots ,k\}$, then ${\left(x_k\right)}_k$ is functionally bounded. If the same holds for $X=X_0$, we call X strongly web-bounding. The family $\{A_{\alpha }:\alpha \in \mathsf{\Omega }\}$ is called, respectively, a web-bounding representation or a strongly web-bounding representation of X.

A topological space X is called a Lindelöf $\mathsf{\Sigma }$-space [6] (or a K-countably determined space [7]) if there is an upper semi-continuous compact-valued map from a non-empty subset $\mathsf{\Omega }\subset N^{\mathbb{N}}$ covering X. If the same holds for $\mathsf{\Omega }={\mathbb{N}}^{\mathbb{N}}$, then X is called K-analytic. X is quasi-Suslin if there exists a set-valued map T from ${\mathbb{N}}^{\mathbb{N}}$ into X covering X which is quasi-Suslin, i.e., if ${\alpha }_n\to \alpha$ in ${\mathbb{N}}^{\mathbb{N}}$ and $x_n\in T\left({\alpha }_n\right)$, then ${\left(x_n\right)}_n$ has a cluster point in $T\left(\alpha \right)$, see [8].

A topological space X is analytic if it is a continuous image of the space ${\mathbb{N}}^{\mathbb{N}}$. Note that analytic ⇒ K-analytic ⇔ Lindelöf ∧ quasi-Suslin, and K-analytic ⇒ Lindelöf $\mathsf{\Sigma }$. Every K-analytic space has a compact resolution, see [9], or [10], and every angelic space with a compact resolution is K-analytic, see [10] (Corollary 1.1).

Recall that topological spaces containing dense quasi-Suslin spaces are web-bounding [5]. Hence, every space containing a dense $\sigma$-compact space is web-bounding, in particular, separable spaces are web-bounding. Applying [1] (Theorem 1, Note 4) we have that a metrizable space is web-bounding if and only if it is separable. Additional information concerning K-analytic properties on spaces $C^b\left(X\right)$ and properties of weakly compact sets in $C\left(X\right)$ are developed in [11,12].

2. Main Results

The following theorems are the main results of this paper that provide two natural characterizations of analyticity. Theorem 1 characterizes when a non-empty subset Y of a locally convex space E in class $\mathfrak{G}$ is $\sigma (E,E^{\prime })$-analytic and Theorem 3 characterizes when non-empty set $Y\subset C_p\left(X\right)$ is analytic, being X a web-bounding space. Although spaces $C_p\left(X\right)$ of continuous real-valued maps on X endowed with the pointwise topology ${\tau }_p$ do not belong to class $\mathfrak{G}$ for uncountable spaces X (as we have mentioned above), the argument used in the proof of Theorem 3 applies to show the general Theorem 1.

Theorem 1.

A subset Y of a locally convex space E in class $\mathfrak{G}$ is $\sigma (E,E^{\prime })$-analytic if and only if Y has a $\sigma (E,E^{\prime })$-compact resolution and is contained in a $\sigma (E,E^{\prime })$-separable subset.

Consequently, a locally convex space E in class $\mathfrak{G}$ is weakly analytic if and only if E is separable and admits a $\sigma (E,E^{\prime })$-compact resolution. Note that the latter condition is equivalent to say that E is weakly K-analytic (since E is angelic by [1] (Theorem 11) and we apply [10] (Corollary 1.1)).

We prove that $C_p\left(X\right)$ is analytic if and only if $C_p\left(X\right)$ has a compact resolution and is separable, see Corollary 2.

Since every analytic compact set is metrizable [1] (Theorem 15), Theorem 1 yields the following result from [2].

Corollary 1

(Cascales-Orihuela). A $\sigma (E,E^{\prime })$-compact set Y in a locally convex space E in class $\mathfrak{G}$ is $\sigma (E,E^{\prime })$-metrizable if and only if Y is contained in a $\sigma (E,E^{\prime })$-separable subset of E.

Moreover, we provide a short proof of the following another interesting result of this type due to Cascales and Orihuela [1].

Theorem 2

(Cascales-Orihuela). A precompact set K in a locally convex space E in class $\mathfrak{G}$ is metrizable.

The following result uses some ideas from [1].

Theorem 3.

Let X be a web-bounding space. A non-empty set $Y\subset C_p\left(X\right)$ is analytic if and only if Y has a compact resolution and is contained in a separable subset of $C_p\left(X\right)$.

3. Examples

Example 1.

In ${\mathbb{R}}^{\mathbb{N}}$ endowed with the product topology, let E be the subspace of ${\mathbb{R}}^{\mathbb{N}}$ formed by the vectors with a finite number of non-null components. Every non-void closed subset Y of E is $\sigma \left(E,E^{\prime }\right)$-analytic.

Proof. 

It is clear that the countable product ${\mathbb{R}}^{\mathbb{N}}$ belongs to class $\mathfrak{G}$, hence E is also in class $\mathfrak{G}$. Let y be an element of Y. For each $\alpha =\left({\alpha }_i:i\in \mathbb{N}\right)\in {\mathbb{N}}^{\mathbb{N}}$ let

$$A_{\alpha }:=\left\{y\right\}\cup \left\{\left(n_i:i\in \mathbb{N}\right)\in Y,n_i=0\mathrm{if}i>{\alpha }_1\right\}.$$

The family $\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ is a compact resolution of Y. Moreover Y is separable, because the topology of ${\mathbb{R}}^{\mathbb{N}}$ has a countable base. By Theorem 1, Y is $\sigma (E,E^{\prime })$-analytic. □

Theorem 2 is Theorem 2 in [1], where the authors provide a picture of possible applications of this Theorem, with detailed proofs concerning that:

• the inductive limits of increasing sequences of metrizable locally convex spaces;
• the generalized inductive limits

  $$E\left[\mathcal{T}\right]=\underset{⟶}{lim}(E_n\left[{\mathcal{T}}_n\right],A_n)$$

  of sequences of pairs $\left\{\right(E_n\left[{\mathcal{T}}_n\right],$$A_n):n=1,2,\dots \}$, where every $A_n$ is ${\mathcal{T}}_n$-metrizable and every $E_n\left[{\mathcal{T}}_n\right]$ is locally convex;
• the locally convex $\left(DF\right)$-spaces;
• and the locally convex dual metric spaces;

are in class $\mathfrak{G}$, hence, its precompact spaces are metrizable:

Example 2.

Let X be the set $\mathbb{N}$ of natural number endowed with the discrete topology. A non-empty set $Y\subset C_p\left(X\right)$ is analytic if and only if Y has a compact resolution.

Proof. 

The space X admits a compact resolution that it is a strongly web-bounding representation of X, hence the space X is strongly web-bounding. If Y has a compact resolution then the Theorem 3 implies that Y is analytic, because the isomorphism between ${\mathbb{R}}^{\mathbb{N}}$ and $C_p\left(X\right)$ implies that $C_p\left(X\right)$ has a countable base. The converse is obvious because analytic $\Rightarrow K$-analytic and every K-analytic space admits a compact resolution. □

4. Proofs

We need the following result [9].

Proposition 1

(Talagrand). Let $(X,\xi )$ be a regular space which admits a stronger topology ϑ such that $(X,\vartheta )$ is a Lindelöf Σ-space. Then $d(X,\vartheta )\le \omega (X,\xi )$, where $d\left(X\right)$ and $\omega \left(X\right)$ denote the density and the weight of X, respectively.

4.1. Proof of Theorem 3

Let us prove Theorem 3.

It is obvious that if Y is analytic then Y is separable and K-analytic, so Theorem 3 holds.

To prove the converse of the statement of this theorem it is enough to show that Y admits a weaker metrizable topology because then, by [1] (Theorem 15), the space Y is analytic.

Firstly we are going to check that to prove this converse we may suppose the additional condition that X is a strongly web-bounding space.

In fact, let X be a web-bounding space and suppose that there is a web-bounding representation $\left\{A_{\alpha }:\alpha \in \mathsf{\Omega }\right\}$ of X whose union $X_0$ is dense in X. Then the restriction map $\varphi :C_p\left(X\right)\to C_p\left(X_0\right)$ defined by ${\varphi \left(f\right):=f|}_{X_0}$ is an injective continuous linear map. Let $Y\subset C_p\left(X\right)$ be a subset with a compact resolution contained in a separable subset $L\subset C_p\left(X\right)$. Then for $\varphi \left(Y\right)$ the assumptions are satisfied, so $\varphi \left(Y\right)$ is analytic in the induced topology from $C_p\left(X_0\right)$ and consequently $\varphi \left(Y\right)$ admits a weaker metrizable topology $\mathcal{T}$. Then $\left\{{\varphi }^{-1}\left(A\right):A\in \mathcal{T}\right\}$ is a weaker metrizable topology on Y. Therefore we may assume that X is strongly web-bounding.

Hence, to finish the proof of Theorem 3 it is enough to prove the following Proposition.

Proposition 2.

Let X be a strongly web-bounding space and let Y be a non-empty subset of $C_p\left(X\right)$ such that Y has a compact resolution and is contained in a separable subset of $C_p\left(X\right)$. Then Y admits a weaker metrizable topology (hence, as was said before, Y is analytic).

Proof. 

Let $\upsilon X$ be the real-compactification of X. Since X is strongly web-bounding, we apply [3] (Theorem 9.15) to deduce that $\upsilon X$ is Lindelöf $\mathsf{\Sigma }$-space.

As a help to the reader we split the proof in two parts.

Step 1. Assume that Y is a subset of $C_p\left(\upsilon X\right)$, Y has a compact resolution and it is contained in a separable subset $L\subset C_p\left(\upsilon X\right)$. Now we prove that L (and also Y) admits a weaker metrizable topology. Let D be a countable dense subset of L. Let ${\mathcal{T}}_D$ and ${\mathcal{T}}_L$ be the weakest topologies on $\upsilon X$ that make continuous the functions of D and L, respectively. By density $f\left(x\right)=f\left(y\right)$ for each $f\in D$ implies $f\left(x\right)=f\left(y\right)$ for each $f\in L$, hence the topological quotients $(\widehat{\upsilon X},\widehat{{\mathcal{T}}_D})$ and $(\widehat{\upsilon X},\widehat{{\mathcal{T}}_L})$ of $(\upsilon X,{\mathcal{T}}_D)$ and $(\upsilon X,{\mathcal{T}}_L)$ respect to the relations $x\sim y$ if $f\left(x\right)=f\left(y\right)$ for all f of D and $x\sim y$ if $f\left(x\right)=f\left(y\right)$ for all f of L, respectively, are algebraically identical and we denote by $\phi :\upsilon X\to \widehat{\upsilon X}$ is the quotient map.

If we define the map $F:\left(\upsilon X,{\mathcal{T}}_D\right)\to {\mathbb{R}}^D$ by $F\left(z\right)=\{f\left(z\right):f\in D\}$, $z\in \upsilon X$, then clearly F is continuous and $x\sim y$ if and only $F\left(x\right)=F\left(y\right)$. $(\widehat{\upsilon X},\widehat{{\mathcal{T}}_D})$ is homeomorphic to a subspace of ${\mathbb{R}}^D$ and consequently $(\widehat{\upsilon X},\widehat{{\mathcal{T}}_D})$ is metrizable and separable. On the other hand $(\widehat{\upsilon X},\widehat{{\mathcal{T}}_L})$ is a Lindelöf $\mathsf{\Sigma }$-space, since it is a continuous image of the Lindelöf $\mathsf{\Sigma }$-space $\upsilon X$. It follows from Proposition 1 that the space $(\widehat{\upsilon X},\widehat{{\mathcal{T}}_L})$ is separable.

Let $S=\{x_n:n\in \mathbb{N}\}$ be a countable subset of $\upsilon X$ such that the set $\phi \left(S\right)$ is $\widehat{{\mathcal{T}}_L}$ dense in $\widehat{\upsilon X}$. For each $f\in L$ let $\widehat{f}$ be the element of $C_p\left(\widehat{\upsilon X}\right)$ such that $f=\widehat{f}\phi$. Let $f,g\in L$ be such that ${f|}_S{=g|}_S$. Then, from $\widehat{f}\phi {{|}_S=\widehat{g}\phi |}_S$ if follows that $\widehat{f}{{|}_{\phi \left(S\right)}=\widehat{g}|}_{\phi \left(S\right)}$ and the density condition implies that $\widehat{f}=\widehat{g}$. Therefore $f=\widehat{f}\phi =\widehat{g}\phi =g$. Consequently, if f and g are two different elements of L there exists $m\in \mathbb{N}$ such that $f\left(x_m\right)\ne g\left(x_m\right)$. This means that the weaker topology on L defined by the topology of the pointwise convergence on S is metrizable.

Step 2. Let $Y\subset C_p\left(X\right)$ be equipped with a compact resolution and let L be a separable set in $C_p\left(X\right)$ containing $Y.$ Let $\psi :C_p\left(X\right)\to C_p\left(\upsilon X\right)$ be defined by $\psi \left(f\right)=f^{\upsilon }$ where $f^{\upsilon }$ is the unique continuous extension of f to the whole $\upsilon X$. Since $\psi$ is continuous on each countable set, see [13] (Theorem 4.6(3)), $\psi \left(Y\right)$ has a resolution of countably compact sets. On the other hand, the space $C_p\left(\upsilon X\right)$ is angelic, see [5] (Theorem 3), so every countably compact set in $C_p\left(\upsilon X\right)$ is compact. Hence, $\psi \left(Y\right)$ has a compact resolution.

Let $\{f_n:n\in \mathbb{N}\}$ be a dense subset of L. Take any $ϵ>0$, any $f^{\upsilon }\in \psi \left(L\right)$ and let $U=\{u_1,u_2,\cdots ,u_p\}$ be an arbitrary finite subset of $\upsilon X$. Then there is $f\in L$ with $\psi \left(f\right)=f^{\upsilon }$ and by [13] (Theorem 4.6(1)) for each $u_i\in U$ there exists $x_i\in X$ such that $f\left(x_i\right)=f^{\upsilon }\left(u_i\right)$ and $f_n\left(x_i\right)=f_n^{\upsilon }\left(u_i\right)$ for each $n\in \mathbb{N}$. Choose $m\in \mathbb{N}$ such that $\left|f_m\left(x_i\right)-f\left(x_i\right)\right|<ϵ$ for each $1\le i\le p$. Hence,

$$\left|f_m^{\upsilon }\left(u_i\right)-f^{\upsilon }\left(u_i\right)\right|=\left|f_m\left(x_i\right)-f\left(x_i\right)\right|<ϵ$$

for each $1\le i\le p$. This shows that $\{f_n^{\upsilon }:n\in \mathbb{N}\}$ is a dense subset of $\psi \left(L\right)$, so that $\psi \left(L\right)$ is separable. By Step 1 we derive that $\psi \left(Y\right)$ is analytic in $C_p\left(\upsilon X\right)$. The continuity of the surjection ${\psi }^{-1}:C_p\left(\upsilon X\right)\to C_p\left(X\right)$ implies that ${\psi }^{-1}\left(\psi \left(Y\right)\right)=Y$ is also analytic. □

For a completely regular topological space X, Tkachuk proved in [14] that $C_p\left(X\right)$ is K-analytic if and only if it has a compact resolution. If X is a separable metric space, then $C_p\left(X\right)$ is analytic if and only if it admits a resolution consisting of bounded sets, see [15] (Corollary 2.5) and [16] (Proposition 1).

From the proof of Proposition 2 follows immediately the following claim that enables to get in Corollary 2 the following variant for analyticity of $C_p\left(X\right)$ for arbitrary X.

Claim 1.

Let X be a topological space such that its real compactification $\upsilon X$ is Lindelöf Σ-space and let Y be a non-empty subset of $C_p\left(X\right)$ such that Y has a compact resolution and is contained in a separable subset of $C_p\left(X\right)$. Then, Y admits a weaker metrizable topology (hence, as was said before, Y is analytic).

Corollary 2.

Let ξ be a topology on $C\left(X\right)$ which is stronger or equal than the pointwise topology ${\tau }_p$ of $C\left(X\right)$. Then $\left(C\right(X),\xi )$ is analytic if and only if $\left(C\right(X),\xi )$ is separable and has a ξ-compact resolution.

Proof. 

It is enough to prove this Corollary when $\xi ={\tau }_p$, because a submetrizable topological space is analytic if and only if it admits a compact resolution (see [1] (Theorem 15)). Assume that $C_p\left(X\right)$ is separable and has a compact resolution. Then by [17] (Corollary 23) the space $\upsilon X$ is a Lindelöf $\mathsf{\Sigma }$-space. Now, Claim 1 for $Y=C_p\left(X\right)$ implies that $C_p\left(X\right)$ is analytic. The converse is clear. □

Hence, a separable space $C_p\left(X\right)$ admits a compact resolution if and only if it is analytic, or, equivalently, there is an upper semi-continuous compact-valued map from ${\mathbb{N}}^{\mathbb{N}}$ covering $C_p\left(X\right)$ if and only if $C_p\left(X\right)$ is a continuous image of ${\mathbb{N}}^{\mathbb{N}}$.

The following example shows that Corollary 2 does not work in general for the weak*-dual $L_p\left(X\right)$ of $C_p\left(X\right)$.

Example 3.

Corollary 2 fails for the weak*-dual $L_p\left({[0,1]}^{\mathbb{R}}\right)$ of $C_p\left({[0,1]}^{\mathbb{R}}\right)$.

Proof. 

It is well known that the space ${[0,1]}^{\mathbb{R}}$ endowed with the product topology is K-analytic separable but not analytic. Consequently $L_p\left({[0,1]}^{\mathbb{R}}\right)$ is K-analytic and separable by [6] (Proposition 0.5.14). $L_p\left({[0,1]}^{\mathbb{R}}\right)$ is not analytic, since ${[0,1]}^{\mathbb{R}}$ is a closed subspace of $L_p\left({[0,1]}^{\mathbb{R}}\right)$ and each closed subspace of an analytic space is analytic. □

4.2. Proofs of Theorems 1 and 2

We are ready to prove Theorem 1.

Proof. 

Note that $(E^{\prime },\sigma (E^{\prime },E))$ is strongly web-bounding. Indeed, let $\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\}$ be a $\mathfrak{G}$-representation of $E^{\prime }$. Then if $\alpha =\left(n_k\right)\in {\mathbb{N}}^{\mathbb{N}}$ and $x_k\in C_{n_1,n_2,\dots ,n_k}$, $k\in \mathbb{N}$, there exists for each $k\in \mathbb{N}$ a ${\beta }_k\in {\mathbb{N}}^{\mathbb{N}}$ such that $x_k\in A_{{\beta }_k}$ and $({\beta }_{k1},{\beta }_{k2},\cdots ,{\beta }_{kk})=(n_1,n_2,\cdots ,n_k)$. From these equalities for $k\in \mathbb{N}$ it follows that there exists $\gamma \in {\mathbb{N}}^{\mathbb{N}}$ with ${\beta }_k\le \gamma$. Hence, $x_k\in A_{\gamma }$ for all $k\in \mathbb{N}$, yielding equicontinuity of ${\left(x_k\right)}_k$, so ${\left(x_k\right)}_k$ is functionally bounded. Finally, as $(E,\sigma (E,E^{\prime }))$ is contained in $C_p(E^{\prime },\sigma (E^{\prime },E))$ the proof follows applying Theorem 3. □

We complete the paper with a short and elementary proof of Theorem 2. It is enough to make the proof for a compact subset K of E, because the completion of a locally convex space E in class $\mathfrak{G}$ belongs to class $\mathfrak{G}$ and the closure in the completion of a precompact subset of E is a compact subset.

Proof. 

Let $\left\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\right\}$ be a $\mathfrak{G}$-representation of $E^{\prime }$. By $\tau$ we denote the topology of E and let K be a compact of $E.$ We say that a subset M of $E^{\prime }$ is $K^0$-separated if $\left(a+K^0\right)\cap M=\left\{a\right\}$, for each $a\in M$. By Zorn’s lemma there exists a maximal $K^0$-separated subset $M_1$ of $E^{\prime }$ and the maximal condition implies that $M_1+K^0=E^{\prime }$.

Note that $M_1$ is countable. Indeed, otherwise, since $E^{\prime }=\left\{A_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}\right\}$ and $A_{\alpha }\subset A_{\beta }$ whenever $\alpha \le \beta$, for $\alpha$, $\beta$ in ${\mathbb{N}}^{\mathbb{N}}$, we determine a sequence $\alpha =\left(n_k\right)\in {\mathbb{N}}^{\mathbb{N}}$ such that each $C_{n_1,n_2,\dots ,n_k}$, $k\in \mathbb{N}$, contains and uncountable subset of $M_1$ and then by a very easy standard argument we obtain countable infinite subset P of $M_1$ and $\gamma \in {\mathbb{N}}^{\mathbb{N}}$ such that $P\subset A_{\gamma }$, see [3,10,18].

Since E belongs to $\mathfrak{G},$ P is equicontinuous, so, by Grothendieck theorem of polar topologies ([19] (Chapter IV, $21.7)) P is precompact in the topology of uniform convergence on the $\tau$-precompact subsets of E. Therefore there exists a finite set $\left\{a_i:1\le i\le k\right\}\subset P$ such that $P\subset \bigcup \left\{a_i+K^0:1\le i\le k\right\}$. Clearly there exists $1\le j\le k$ such that the set $\left(a_j+K^0\right)\cap P$ is infinite, contradicting the hypothesis that $M_1$ ($\supset P$) is $K^0$-separated.

Let $M_n$ be a maximal subset of $E^{\prime }$ that it is $n^{-1}{K}^0$-separated, for each $n\in \mathbb{N}$. The set $M_0:=\bigcup \left\{M_n:n\in \mathbb{N}\right\}$ is countable. Let ${\tau }_{M_0}$ be the weakest topology on K that makes continuous the functions of $M_0$. If $x\ne y$ are two points of K then there exist $g\in E^{\prime }$ and $n\in \mathbb{N}$ such that $\left|g\left(x\right)-g\left(y\right)\right|>3n^{-1}$. Since $E^{\prime }=M_n+n^{-1}{K}^0$, there exists $f\in M_n(\subset M_0)$ such that $g\in f+n^{-1}{K}^0$. Hence,

$$\left|f\left(x\right)-f\left(y\right)\right|=\left|g\left(x\right)-g\left(y\right)-g\left(x\right)+f\left(x\right)+g\left(y\right)-f\left(y\right)\right|>3n^{-1}-2n^{-1}=n^{-1}.$$

Therefore $(K,{\tau }_{M_0})$ is metrizable, so K is metrizable. □

5. Conclusions

For a locally convex space E in class $\mathfrak{G}$, we have characterized that a subset Y of $(E,\sigma (E,E^{\prime }))$, endowed with the induced topology, is $\sigma (E,E^{\prime })$-analytic if and only if Y has a $\sigma (E,E^{\prime })$-compact resolution and is contained in a $\sigma (E,E^{\prime })$-separable subset of E. If X is a web-bounding space, then we have obtained that a non-empty subset Y of $C_p\left(X\right)$ provided with the induced topology is analytic if and only if Y has a compact resolution and is contained in a separable subset of $C_p\left(X\right)$. Moreover, for a topology $\xi$ on $C\left(X\right)$ which is stronger or equal to the pointwise topology ${\tau }_p$ of $C\left(X\right)$ we obtain that $\left(C\right(X),\xi )$ is analytic if and only if $\left(C\right(X),\xi )$ is separable and has a $\xi$-compact resolution. This last result suggests for future work to characterize the locally convex spaces E in class $\mathfrak{G}$ that are analytic, being $\xi$ a topology stronger than the weak topology $\sigma (E,E^{\prime })$.

Another direction of future research is to obtain similar characterizations for spaces in class $\mathfrak{G}$ and for spaces $C_p\left(X\right)$ replacing analytic by weaker properties like to be K-analytic or quasi-Suslin.