Metrizable-like locally convex topologies on C(X)

J. C. Ferrando; S. Gabriyelyan; J. Ka̧kol

doi:10.1016/j.topol.2017.07.016

Metrizable-like locally convex topologies on C(X)

J. C. Ferrando, S. Gabriyelyan, J. Ka̧kol

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

The classic Arens theorem states that the space C(X) of real-valued continuous functions on a Tychonoff space X is metrizable in the compact-open topology τ_k if and only if X is hemicompact. Less demanding but still applicable problem asks whether τ_k has an N^N-decreasing base at zero (U_α)_α∈N^N, called in the literature a G-base. We characterize those spaces X for which C(X) admits a locally convex topology T between the pointwise topology τ_p and the bounded-open topology τ_b such that (C(X),T) is either metrizable or is an (LM)-space or even has a G-base.

Original language	English
Pages (from-to)	105-113
Number of pages	9
Journal	Topology and its Applications
Volume	230
DOIs	https://doi.org/10.1016/j.topol.2017.07.016
State	Published - 1 Oct 2017

Keywords

(LM)-topology
Functionally bounded set
G-base
Hewitt realcompactification
K-analytic
Metrizable

ASJC Scopus subject areas

Geometry and Topology

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10.1016/j.topol.2017.07.016

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title = "Metrizable-like locally convex topologies on C(X)",

abstract = "The classic Arens theorem states that the space C(X) of real-valued continuous functions on a Tychonoff space X is metrizable in the compact-open topology τk if and only if X is hemicompact. Less demanding but still applicable problem asks whether τk has an NN-decreasing base at zero (Uα)α∈NN, called in the literature a G-base. We characterize those spaces X for which C(X) admits a locally convex topology T between the pointwise topology τp and the bounded-open topology τb such that (C(X),T) is either metrizable or is an (LM)-space or even has a G-base.",

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AU - Gabriyelyan, S.

AU - Ka̧kol, J.

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AB - The classic Arens theorem states that the space C(X) of real-valued continuous functions on a Tychonoff space X is metrizable in the compact-open topology τk if and only if X is hemicompact. Less demanding but still applicable problem asks whether τk has an NN-decreasing base at zero (Uα)α∈NN, called in the literature a G-base. We characterize those spaces X for which C(X) admits a locally convex topology T between the pointwise topology τp and the bounded-open topology τb such that (C(X),T) is either metrizable or is an (LM)-space or even has a G-base.

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KW - Functionally bounded set

KW - G-base

KW - Hewitt realcompactification

KW - K-analytic

KW - Metrizable

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JO - Topology and its Applications

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ER -

Metrizable-like locally convex topologies on C(X)

Abstract

Keywords

ASJC Scopus subject areas

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