The $\kappa$-Fr\'{e}chet--Urysohn property for locally convex spaces

TL;DR

This paper investigates the $ppa$-Fre9chet--Urysohn property in topological spaces, showing it implies the Ascoli property and characterizing this in locally convex spaces, especially for spaces of continuous functions.

Contribution

It establishes that $ppa$-Fre9chet--Urysohn spaces are Ascoli and characterizes this property in various locally convex spaces, answering a specific open question.

Findings

Every $ppa$-Fre9chet--Urysohn Tychonoff space is Ascoli.

Characterization of the $ppa$-Fre9chet--Urysohn property in locally convex spaces.

$C_p(X)$ is Ascoli iff $X$ has property $(ppa)$.

Abstract

A topological space $X$ is $\kappa$-Fr\'{e}chet--Urysohn if for every open subset $U$ of $X$ and every $x\in \overline{U}$ there exists a sequence in $ U$ converging to $x$. We prove that every $\kappa$-Fr\'{e}chet--Urysohn Tychonoff space $X$ is Ascoli. We apply this statement and some of known results to characterize the $\kappa$-Fr\'echet--Urysohn property in various important classes of locally convex spaces. In particular, answering a question posed in [7] we obtain that $C_p(X)$ is Ascoli iff $X$ has the property $(\kappa)$.

Equations

$$\sigma(z):=\big{\{}x=(x_{i})\in X:\{i:x_{i}\not=z_{i}\}\mbox{ is finite}\big{\}}.$$

$$W:=\bigcup \{W_{f_U}:U\text{mbox}isanopenneighborhoodofz\}.$$

$$|f_{U_n}(x_n)-f_{U_n}(z)|>{\epsilon }_0(\text{mbox}foralln\in \mathbb{N}).$$

$$\big{|}f_{U_{n}}(x_{i})-f_{U_{n}}(z)\big{|}<\frac{\varepsilon_{0}}{2}\;\;\mbox{ for all }\,i\geq N\,\mbox{ and }\,n\in\mathbb{N}.$$

$$\{f\in \phi :|f(n)|<{\epsilon }_n\text{mbox}foreveryn\in \mathbb{N}\},$$

$$U_{n,k}:=\left\{f\in \phi :|f(1)|>\frac{1}{2n}\text{mbox}and|f(n)|>\frac{1}{2k}\right\},$$

$$V:=\{f\in \phi :|f(n)|<{\epsilon }_n\text{mbox}foreveryn\in \mathbb{N}\}.$$

$$E_0\hookrightarrow E_1\hookrightarrow E_2\hookrightarrow \cdots$$

Full text

1 Introduction

Following Arhangel’skii, a topological space $X$ is said to be $\kappa$-Fréchet–Urysohn if for every open subset $U$ of $X$ and every $x\in\overline{U}$, there exists a sequence $\{x_{n}\}_{n\in\mathbb{N}}\subseteq U$ converging to $x$. Clearly, every Fréchet–Urysohn space is $\kappa$-Fréchet–Urysohn. In [10, Theorem 3.3] Liu and Ludwig showed that a topological space $X$ is $\kappa$-Fréchet–Urysohn if and only if $X$ is a $\kappa$-pseudo open image of a metric space. Below we give another characterization of $\kappa$-Fréchet–Urysohn spaces, see 2.1. It is known that there are $\kappa$-Fréchet–Urysohn spaces which are not $k$-spaces, and there are sequential spaces which are not $\kappa$-Fréchet–Urysohn, see [10] or Proposition 2.6 below.

Let $X$ be a Tychonoff (=completely refular and Hausdorff) space. Denote by $C_{k}(X)$ and $C_{p}(X)$ the space $C(X)$ of all real-valued continuous functions on $X$ endowed with the compact-open topology and the pointwise topology, respectively. Following [2], $X$ is called an Ascoli space if every compact subset $\mathcal{K}$ of $C_{k}(X)$ is evenly continuous (i.e., if the map $(f,x)\mapsto f(x)$ is continuous as a map from $\mathcal{K}\times X$ to $\mathbb{R}$). In [4] we noticed that $X$ is Ascoli if and only if every compact subset of $C_{k}(X)$ is equicontinuous. The classical Ascoli theorem [3, Theorem 3.4.20] states that every $k$-space is Ascoli.

In [14, Theorem 2.1], Sakai characterized those spaces $C_{p}(X)$ which are $\kappa$-Fréchet–Urysohn. Recall that a family $\{A_{i}\}_{i\in I}$ of subsets of a set $X$ is said to be point-finite if the set $\{i\in I:x\in A_{i}\}$ is finite for every $x\in X$. A family $\{A_{i}\}_{i\in I}$ of subsets of a topological space $X$ is called strongly point-finite if for every $i\in I$, there exists an open set $U_{i}$ of $X$ such that $A_{i}\subseteq U_{i}$ and $\{U_{i}\}_{i\in I}$ is point-finite. Following Sakai [14], a topological space $X$ is said to have the property $(\kappa)$ if every sequence of pairwise disjoint finite subsets of $X$ has a strongly point-finite subsequence.

Theorem 1.1** ([14])**

The space $C_{p}(X)$ is $\kappa$-Fréchet–Urysohn if and only if $X$ has the property $(\kappa)$.

A characterization of the spaces $C_{k}(X)$ which are $\kappa$-Fréchet–Urysohn is given in [15].

In [7] we proved the following theorem.

Theorem 1.2** ([7])**

If $C_{p}(X)$ is Ascoli, then it is $\kappa$-Fréchet–Urysohn.

However, the question (see [7, Question 2.4]) of whether every $\kappa$-Fréchet–Urysohn space $C_{p}(X)$ is Ascoli remained open. In this short note we answer this question in the affirmative using the following somewhat unexpected result.

Theorem 1.3

Each $\kappa$-Fréchet–Urysohn space $X$ is Ascoli.

Now Theorems 1.1-1.3 immediately imply the following characterization of spaces $C_{p}(X)$ which are Ascoli.

Corollary 1.4

Let $X$ be a Tychonoff space. Then $C_{p}(X)$ is Ascoli if and only if $X$ has the property $(\kappa)$.

Denote by $\mathcal{D}(\Omega)$ the space of test functions over an open subset $\Omega$ of $\mathbb{R}^{n}$. In [5] we proved that $\mathcal{D}(\Omega)$ and the strong dual $\mathcal{D}^{\prime}(\Omega)$ of $\mathcal{D}(\Omega)$, the space of distributions, are not Ascoli. Therefore, by Theorem 1.3, $\mathcal{D}(\Omega)$ and $\mathcal{D}^{\prime}(\Omega)$ are not $\kappa$-Fréchet–Urysohn spaces. Below we apply Theorem 1.3 and some of the main results from [1, 4, 5, 6, 8] to characterize the $\kappa$-Fréchet–Urysohness in various important classes of locally convex spaces.

2 Proof of Theorem 1.3

We start from the following characterization of $\kappa$-Fréchet–Urysohn spaces. The closure of a subset $A$ of a topological space $X$ is denoted by $\overline{A}$ or $\mathrm{cl}_{X}(A)$.

Theorem 2.1

A topological space $X$ is $\kappa$-Fréchet–Urysohn if and only if each point $x\in X$ is contained in a dense $\kappa$-Fréchet–Urysohn subspace of $X$.

Proof 1

The necessity is clear. To prove sufficiency, fix an open subset $U$ of $X$ and a point $x\in\overline{U}$. Let $Y$ be a dense $\kappa$-Fréchet–Urysohn subspace of $X$ containing $x$. Then $V:=U\cap Y$ is an open subset of $Y$. We claim that $x\in\mathrm{cl}_{Y}(V)$. Indeed, if $W\subseteq Y$ is an open neighborhood of $x$ in $Y$, take an open $W^{\prime}\subseteq X$ such that $W=W^{\prime}\cap Y$. Then the set $W^{\prime}\cap U$ is open in $X$. Since $Y$ is dense in $X$ the set $(W^{\prime}\cap U)\cap Y=(W^{\prime}\cap Y)\cap(U\cap Y)=W\cap V$ is not empty. Thus $x\in\mathrm{cl}_{Y}(V)$ and the claim is proved. Finally, since $Y$ is $\kappa$-Fréchet–Urysohn there is a sequence $\{y_{n}\}_{n\in\mathbb{N}}\subseteq V\subseteq U$ converging to $x$. \qed

Corollary 2.2

Let $Y$ be a dense subset of a homogeneous space (in particular, a topological group) $X$. If $Y$ is $\kappa$-Fréchet–Urysohn, then $X$ is also a $\kappa$-Fréchet–Urysohn.

Proof 2

Fix arbitrarily $y_{0}\in Y$. Let $x\in X$. Take a homeomorphism $h$ of $X$ such that $h(y_{0})=x$. Then $x\in h(Y)$ and $h(Y)$ is a $\kappa$-Fréchet–Urysohn space. Therefore, each element of $X$ is contained in a dense $\kappa$-Fréchet–Urysohn subspace of $X$ and Theorem 2.1 applies. \qed

In [10, Theorem 4.1] Liu and Ludwig proved that the product of a family of bi-sequential spaces is $\kappa$-Fréchet–Urysohn. Note that any countable product of bi-sequential spaces is bi-sequential, see [12, Proposition 3.D.3]. On the other hand, countable products of $W$-spaces are $W$-spaces ([9, Theorem 4.1]) and there are $W$-spaces which are not bi-sequential ([9, Example 5.1]). Taking into account that bi-sequential spaces and $W$-spaces are Fréchet–Urysohn spaces, the next corollary essentially generalizes Theorem 4.1 of [10].

Corollary 2.3

Let $\{X_{i}:i\in I\}$ be a family of topological spaces such that $\prod_{i\in I^{\prime}}X_{i}$ is Fréchet–Urysohn for any countable subset $I^{\prime}$ of $I$. Then the space $X=\prod_{i\in I}X_{i}$ is $\kappa$-Fréchet–Urysohn.

Proof 3

For every $z=(z_{i})\in X$, set

[TABLE]

Clearly, $\sigma(z)$ is a dense subspace of $X$. Proposition 2.6 of [7] states that $\sigma(z)$ is Fréchet–Urysohn. By Theorem 2.1, $X$ is $\kappa$-Fréchet–Urysohn. \qed

Below we prove Theorem 1.3.

Proof of Theorem 1.3. Suppose for a contradiction that $X$ is not an Ascoli space. Then there exists a compact set $K$ in $C_{k}(X)$ which is not equicontinuous at some point $z\in X$. Therefore there is $\varepsilon_{0}>0$ such that for every open neighborhood $U$ of $z$ there exists a function $f_{U}\in K$ for which the open set $W_{f_{U}}:=\{x\in U:|f_{U}(x)-f_{U}(z)|>\varepsilon_{0}\}$ is not empty. Set

[TABLE]

Then $W$ is an open subset of $X$ such that $z\in\overline{W}\setminus W$. As $X$ is $\kappa$-Fréchet–Urysohn, there is a sequence $\{x_{n}:n\in\mathbb{N}\}\subseteq W$ converging to $z$. For every $n\in\mathbb{N}$, choose an open neighborhood $U_{n}$ of $z$ such that $x_{n}\in W_{f_{U_{n}}}(\subseteq U_{n})$ and, therefore,

[TABLE]

Set $S:=\{x_{n}:n\in\mathbb{N}\}\cup\{z\}$. Then $S$ is a compact subset of $X$. Denote by $p$ the restriction map $p:C_{k}(X)\to C_{k}(S),p(f)=f|_{S}$. Then $p(K)$ is a compact subset of the Banach space $C_{k}(S)$. Applying the Ascoli theorem to the compact space $S$ we obtain that the sequence $\{p(f_{U_{n}})\}_{n\in\mathbb{N}}\subseteq p(K)$ is equicontinuous at $z\in S$ and, therefore, there is an $N\in\mathbb{N}$ such that

[TABLE]

In particular, for $i=n=N$ we obtain \big{|}f_{U_{N}}(x_{N})-f_{U_{N}}(z)\big{|}<\frac{\varepsilon_{0}}{2}. But this contradicts (2.1). Thus $X$ is an Ascoli space. \qed

The next corollary strengthens Theorem 1.3 of [7].

Corollary 2.4

Let $X$ be a Čech-complete space. Then $C_{p}(X)$ is Ascoli if and only if $X$ is scattered.

Proof 4

If $C_{p}(X)$ is Ascoli, then $X$ is scattered by Theorem 1.3 of [7]. Conversely, if $X$ is scattered, then, by Corollary 3.8 of [14], $X$ has the property $(\kappa)$. Thus, by Corollary 1.4, $C_{p}(X)$ is Ascoli.\qed

Let $E$ be a locally convex space over a field $\mathbf{F}$, where $\mathbf{F}=\mathbb{R}$ or $\mathbb{C}$, and let $E^{\prime}$ the dual space of $E$. If $E$ is a Banach space, denote by $B$ the closed unit ball of $E$ and set B_{w}:=\big{(}B,\sigma(E,E^{\prime})|_{B}\big{)}, where $\sigma(E,E^{\prime})$ is the weak topology on $E$.

Corollary 2.5

(i)* If $E$ is a Banach space, then $B_{w}$ is $\kappa$-Fréchet–Urysohn if and only if $E$ does not contain an isomorphic copy of $\ell_{1}$.*

(ii)* A Fréchet space $E$ over $\mathbf{F}$ is $\kappa$-Fréchet–Urysohn in the weak topology if and only if $E=\mathbf{F}^{N}$ for some $N\leq\omega$.*

(iii)* If $X$ is a $\mu$-space and a $k_{\mathbb{R}}$-space, then $C_{k}(X)$ is $\kappa$-Fréchet–Urysohn in the weak topology if and only if $X$ is discrete.*

(iv)* The weak∗ dual space of a metrizable barrelled space $E$ is $\kappa$-Fréchet–Urysohn if and only if $E$ is finite-dimensional.*

Proof 5

(i) Theorem 1.9 of [8] or Theorem 6.1.1 and Corollary 1.7 of [1] state that $B_{w}$ is Ascoli if and only if $B_{w}$ is Fréchet–Urysohn if and only if $E$ does not contain an isomorphic copy of $\ell_{1}$. Now Theorem 1.3 applies.

(ii) Corollary 1.7 of [4] states that $E$ is Ascoli in the weak topology if and only if $E=\mathbf{F}^{N}$ for some $N\leq\omega$. This result and Theorem 1.3 imply the desired.

(iii) Corollary 1.9 of [4] states that $C_{k}(X)$ is Ascoli in the weak topology if and only if $X$ is discrete. Now the assertion follows from Theorem 1.3 and the fact that any product of metrizable spaces is $\kappa$-Fréchet–Urysohn (see Fact 1.2 of [10]).

(iv) Corollary 1.14 of [4] states that the weak∗ dual space of $E$ is Ascoli if and only if $E$ is finite-dimensional, and Theorem 1.3 applies. \qed

Now we consider direct locally convex sums of locally convex spaces. The simplest infinite direct sum of lcs is the space $\varphi$, the direct locally convex sum $\bigoplus_{n\in\mathbb{N}}E_{n}$ with $E_{n}=\mathbf{F}$ for all $n\in\mathbb{N}$. It is well known that $\varphi$ is a sequential non-Fréchet–Urysohn space, see Example 1 of [13].

Proposition 2.6

An infinite direct sum of (non-trivial) locally convex spaces is not $\kappa$-Fréchet–Urysohn. In particular, $\varphi$ is not a $\kappa$-Fréchet–Urysohn space.

Proof 6

Let $L=\bigoplus_{i\in I}E_{i}$ be the direct locally convex sum of an infinite family $\{E_{i}\}_{i\in I}$ of locally convex spaces. It is well known that every $E_{i}$ can be represented as a direct sum $\mathbf{F}\oplus E^{\prime}_{i}$. Therefore $L$ contains $\varphi$ as a direct summand. Since the projection of $L$ onto $\varphi$ is open and the $\kappa$-Fréchet–Urysohn property is preserved under open maps (see Proposition 3.3 of [10]), it is sufficient to show that $\varphi$ is not a $\kappa$-Fréchet–Urysohn space.

We consider elements of $\varphi$ as functions from $\mathbb{N}$ to $\mathbf{F}$ with finite support. Recall that the sets of the form

[TABLE]

where $\varepsilon_{n}>0$ for all $n\in\mathbb{N}$, form a basis at [math] of $\varphi$ (see for example [13, Example 1]). For every $n,k\in\mathbb{N}$, set

[TABLE]

and set $U:=\bigcup_{n,k\in\mathbb{N}}U_{n,k}$. It is easy to see that all the sets $U_{n,k}$ are open in $\varphi$ and $0\not\in U_{n,k}$. Hence $U$ is an open subset of $\varphi$ such that $0\not\in U$. To show that $\varphi$ is not $\kappa$-Fréchet–Urysohn, it suffices to prove that (A) $0\in\overline{U}$, and (B) there is no a sequence in $U$ converging to [math].

(A) Let $W$ be a basic neighborhood of zero in $\varphi$ of the form (2.2). Choose an $n\in\mathbb{N}$ such that $\frac{1}{n}<\varepsilon_{1}$, and take $k\in\mathbb{N}$ such that $\frac{1}{k}<\varepsilon_{n}$. It is clear that $U_{n,k}\cap W$ is not empty. Thus $0\in\overline{U}$.

(B) Suppose for a contradiction that there is a sequence $S=\{f_{j}\}_{j\in\mathbb{N}}$ in $U$ converging to [math]. For every $j\in\mathbb{N}$, take $n_{j},k_{j}\in\mathbb{N}$ such that $f_{j}\in U_{n_{j},k_{j}}$. Since $f_{j}\to 0$, the definition of $U_{n,k}$ implies that $\frac{1}{2n_{j}}<|f_{j}(1)|\to 0$, and hence $n_{j}\to\infty$. Without loss of generality we can assume that $1<n_{1}<n_{2}<\cdots$. For every $n\in\mathbb{N}$, define $\varepsilon_{n}=\frac{1}{4k_{j}}$ if $n=n_{j}$ for some $j\in\mathbb{N}$, and $\varepsilon_{n}=1$ otherwise. Set

[TABLE]

Then, $V$ is a neighborhood of [math], and the construction of $U_{n,k}$ implies that $V\cap U_{n_{j},k_{j}}=\emptyset$ for every $j\in\mathbb{N}$. Thus $S\cap V=\emptyset$ and hence $f_{j}\not\to 0$, a contradiction. \qed

Recall that a strict $(LF)$-space $E$ is the direct limit $E=\underrightarrow{\mbox{\rm s-ind}}_{n}\,E_{n}$ of an increasing sequence

[TABLE]

of Fréchet (= locally convex complete metric linear) spaces in the category of locally convex spaces and continuous linear maps. The space $\mathcal{D}(\Omega)$ of test functions is one of the most famous and important examples of strict $(LF)$-spaces.

Corollary 2.7

A strict $(LF)$-space $E$ is $\kappa$-Fréchet–Urysohn if and only if $E$ is a Fréchet space.

Proof 7

Theorem 1.2 of [5] states that $E$ is an Ascoli space if and only if $E$ is a Fréchet space or $E=\varphi$. Now the assertion follows from Theorem 1.3 and Proposition 2.6. \qed

One of the most important classes of locally convex spaces is the class of free locally convex spaces. Following [11], the free locally convex space $L(X)$ on a Tychonoff space $X$ is a pair consisting of a locally convex space $L(X)$ and a continuous map $i:X\to L(X)$ such that every continuous map $f$ from $X$ to a locally convex space $E$ gives rise to a unique continuous linear operator ${\bar{f}}:L(X)\to E$ with $f={\bar{f}}\circ i$. The free locally convex space $L(X)$ always exists and is essentially unique.

Corollary 2.8

Let $X$ be a Tychonoff space. Then $L(X)$ is a $\kappa$-Fréchet–Urysohn space if and only if $X$ is finite.

Proof 8

It is well known that $L(D)$ over a countably infinite discrete space $D$ is topologically isomorphic to $\varphi$. By Theorem 1.2 of [6], $L(X)$ is an Ascoli space if and only if $X$ is a countable discrete space. This fact, Theorem 1.3 and Proposition 2.6 immediately imply the assertion. \qed