On first and second countable spaces and the axiom of choice
type: Item
instance of: scholarly article
title: On first and second countable spaces and the axiom of choice
zbMATH Open document ID: 1065.54015
author: Gonçalo Gutierres
published in: Topology and its Applications
publication date: 2004-10-01T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/4635
review text: This paper discusses the role of the axiom of countable choice for families of sets of real numbers in first and second countable spaces. All spaces are considered in ZF. Denote by \(\text{CC}(\mathbb{R})\) the axiom of countable choice for families of sets of real numbers. The following concept is introduced: A topological space is called super second countable (SSC) if each base has a countable subfamily which is a base. The author proves that the following propositions are equivalent: (1) \(\text{CC}(\mathbb{R})\), (2) the axiom of countable choice is true for families of dense subspaces of \(\mathbb{R}\), (3) every subspace of \(\mathbb{R}\) is separable, (4) every dense subspace of \(\mathbb{R}\) is separable, (5) every base of a second countable space has a countable subfamily which is a base, (6) every base of open sets of \(\mathbb{R}\) has a countable subfamily which is a base, (7) \(\mathbb{R}\) is SSC, (8) every separable metric space is SSC. Some questions are mentioned: (1) Is SSC hereditary? (2) Are there non-separable SSC-spaces? (3) Are there uncountable SSC \(T_0\)-spaces? For the first countable case, two conditions are stated: one of them classical, namely CUT: the countable union of countable sets is countable. Another is a recent statement: \(\omega\)-MC: for each family \((X_i)_{i\in I}\) of non-empty sets, there is a family \((A_i)_{i\in I}\) such that each \(A_i\) is non-empty, finite or countable infinite and \(A_i\subset X_i\) for all \(i\in I\). Then the following are equivalent: (1) \(\omega\)-MC and CC, (2) \(\omega\)-MC and CUT, (3) In a first countable space, for each local base \(({\mathfrak B}(x))_{x\in X}\), there is a local base \(({\mathfrak V}(x))_{x\in X}\), where for each \(x\in X\), \({\mathfrak V}(x)\) is countable, and \({\mathfrak V}(x)\subset{\mathfrak B}(x)\). Moreover, some results are mentioned. If the countable product of second countable spaces is second countable, CUT holds. CC holds for families of sets with at most…
Mathematics Subject Classification ID: 54D70
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 2104475
zbMATH Keywords: CC
zbMATH Keywords: SSC
zbMATH Keywords: \(\omega\)-MC
zbMATH Keywords: first countable space
zbMATH Keywords: second countable space
reviewed by: Kiyoshi Iseki
MaRDI profile type: Publication
OpenAlex ID: W2075457809
Wikidata QID: Q114013001
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DOI: 10.1016/J.TOPOL.2004.02.010
description: scientific article; zbMATH DE number 2104475