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

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

18 Scopus citations

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	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

ASJC Scopus subject areas

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

Access to Document

10.1090/bproc/76

Cite this

@article{48eb3dea42e44a8e814b9d61e2ee011a,

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.",

keywords = "Distinguished locally convex space, Isbell–Mr{\'o}wka space, scattered compact space, Δ-set",

author = "Jerzy Ka¸Kol and Arkady Leiderman",

note = "Publisher Copyright: {\textcopyright} 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)",

year = "2021",

month = jan,

day = "1",

doi = "10.1090/bproc/76",

language = "English",

volume = "8",

pages = "86--99",

journal = "Proceedings of the American Mathematical Society, Series B",

issn = "2330-1511",

publisher = "American Mathematical Society",

}

TY - JOUR

T1 - A CHARACTERIZATION OF X FOR WHICH SPACES Cp(X) ARE DISTINGUISHED AND ITS APPLICATIONS

AU - Ka¸Kol, Jerzy

AU - Leiderman, Arkady

PY - 2021/1/1

Y1 - 2021/1/1

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.

KW - Distinguished locally convex space

KW - Isbell–Mrówka space

KW - scattered compact space

KW - Δ-set

UR - http://www.scopus.com/inward/record.url?scp=85104607611&partnerID=8YFLogxK

U2 - 10.1090/bproc/76

DO - 10.1090/bproc/76

M3 - Article

AN - SCOPUS:85104607611

SN - 2330-1511

VL - 8

SP - 86

EP - 99

JO - Proceedings of the American Mathematical Society, Series B

JF - Proceedings of the American Mathematical Society, Series B

ER -