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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Preliminary Result

**Theorem 1**

- 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

**Lemma 1.**

**Proof.**

**Theorem 2.**

- 1.
- 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.**

**Remark 1.**

**Example 1.**

**Example 2.**

**Example 3.**

**Example 4.**

**Example 5.**

**Example 6.**

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

**Lemma 2.**

**Proof.**

**Lemma 3.**

**Proof.**

**Remark 2.**

**Proof.**

**Definition 1.**

**Theorem 3.**

**Proof.**

## 5. Conclusions

**Theorem 4.**

- 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.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Ferrando, J.C.; López-Pellicer, M.
Classifying Topologies through *Axioms* **2022**, *11*, 744.
https://doi.org/10.3390/axioms11120744

**AMA Style**

Ferrando JC, López-Pellicer M.
Classifying Topologies through *Axioms*. 2022; 11(12):744.
https://doi.org/10.3390/axioms11120744

**Chicago/Turabian Style**

Ferrando, Juan Carlos, and Manuel López-Pellicer.
2022. "Classifying Topologies through *Axioms* 11, no. 12: 744.
https://doi.org/10.3390/axioms11120744