A CHARACTERIZATION OF X FOR WHICH SPACES Cp(X) ARE DISTINGUISHED AND ITS APPLICATIONS

Jerzy Ka¸Kol; Arkady Leiderman

doi:10.1090/bproc/76

A CHARACTERIZATION OF X FOR WHICH SPACES C_p(X) ARE DISTINGUISHED AND ITS APPLICATIONS

Jerzy Ka¸Kol, Arkady Leiderman

Department of Mathematics

Ben-Gurion University of the Negev

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

Abstract

We prove that the locally convex space C_p(X) of continuous real-valued functions on a Tychonoff space X equipped with the topology of point-wise convergence is distinguished if and only if X is a Δ-space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60]. As an application of this characterization theorem we obtain the following results: 1) If X is a Čech-complete (in particular, compact) space such that C_p(X) is distinguished, then X is scattered. 2) For every separable compact space of the Isbell–Mrówka type X, the space C_p(X) is distinguished. 3) If X is the compact space of ordinals [0, ω₁], then C_p(X) is not distinguished. We observe that the existence of an uncountable separable metrizable space X such that C_p(X) is distinguished, is independent of ZFC. We also explore the question to which extent the class of Δ-spaces is invariant under basic topological operations.

Original language	American English
Pages (from-to)	86-99
Number of pages	14
Journal	Proceedings of the American Mathematical Society, Series B
Volume	8
DOIs	https://doi.org/10.1090/bproc/76
State	Published - 1 Jan 2021

Keywords

Distinguished locally convex space
Isbell–Mrówka space
scattered compact space
Δ-set

All Science Journal Classification (ASJC) codes

Analysis
Discrete Mathematics and Combinatorics
Geometry and Topology
Algebra and Number Theory

Access to Document

10.1090/bproc/76

Cite this

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title = "A CHARACTERIZATION OF X FOR WHICH SPACES Cp(X) ARE DISTINGUISHED AND ITS APPLICATIONS",

abstract = "We prove that the locally convex space Cp(X) of continuous real-valued functions on a Tychonoff space X equipped with the topology of point-wise convergence is distinguished if and only if X is a Δ-space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60]. As an application of this characterization theorem we obtain the following results: 1) If X is a {\v C}ech-complete (in particular, compact) space such that Cp(X) is distinguished, then X is scattered. 2) For every separable compact space of the Isbell–Mr{\'o}wka type X, the space Cp(X) is distinguished. 3) If X is the compact space of ordinals [0, ω1], then Cp(X) is not distinguished. We observe that the existence of an uncountable separable metrizable space X such that Cp(X) is distinguished, is independent of ZFC. We also explore the question to which extent the class of Δ-spaces is invariant under basic topological operations.",

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N2 - We prove that the locally convex space Cp(X) of continuous real-valued functions on a Tychonoff space X equipped with the topology of point-wise convergence is distinguished if and only if X is a Δ-space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60]. As an application of this characterization theorem we obtain the following results: 1) If X is a Čech-complete (in particular, compact) space such that Cp(X) is distinguished, then X is scattered. 2) For every separable compact space of the Isbell–Mrówka type X, the space Cp(X) is distinguished. 3) If X is the compact space of ordinals [0, ω1], then Cp(X) is not distinguished. We observe that the existence of an uncountable separable metrizable space X such that Cp(X) is distinguished, is independent of ZFC. We also explore the question to which extent the class of Δ-spaces is invariant under basic topological operations.

AB - We prove that the locally convex space Cp(X) of continuous real-valued functions on a Tychonoff space X equipped with the topology of point-wise convergence is distinguished if and only if X is a Δ-space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60]. As an application of this characterization theorem we obtain the following results: 1) If X is a Čech-complete (in particular, compact) space such that Cp(X) is distinguished, then X is scattered. 2) For every separable compact space of the Isbell–Mrówka type X, the space Cp(X) is distinguished. 3) If X is the compact space of ordinals [0, ω1], then Cp(X) is not distinguished. We observe that the existence of an uncountable separable metrizable space X such that Cp(X) is distinguished, is independent of ZFC. We also explore the question to which extent the class of Δ-spaces is invariant under basic topological operations.

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