Some weaker monotone separation and basis properties
type: Item
instance of: scholarly article
title: Some weaker monotone separation and basis properties
zbMATH Open document ID: 0848.54015
DOI: 10.1016/0166-8641(95)00069-0
author: Robert E. Buck
published in: Topology and its Applications
publication date: 1996-10-28T00:00:00Z
review text: A topological space is called an \(m_1\)-space (respectively an \(m_2\)-space) if each point has a closure preserving local base (respectively a closure preserving local quasi-base). It is called an \(m_3\)-space if each point has a cushioned local pair-base. Obviously, every first countable space is an \(m_1\)-space, and every \(m_1\)-space is an \(m_2\)-space. In [ibid. 19, 65-69 (1985; Zbl 0567.54012)] \textit{M. Ito} has shown that every stratifiable \(m_1\)-space is an \(M_1\)-space. Motivated by Ito's theorem, this paper contains a systematic study of \(m_i\)-spaces. It is shown that every strongly monotonically \(T_2\)-space is an \(m_2\)-space. Every regular \(m_2\)-space is an \(m_3\)-space. However, there exist first countable Hausdorff spaces, hence Hausdorff \(m_2\)-spaces, which are not \(m_3\)-spaces. Every Hausdorff \(m_3\)-space is regular. More precisely, a topological space \(X\) is a Hausdorff \(m_3\)-space if and only if it is monotonically \(T_2\), i.e. if there exists a function \(g\) assigning to each ordered pair \((x,y)\) of distinct points of \(X\) an open neighbourhood \(g(x,y)\) of \(x\) such that (i) \(g(x,y) \cap g(y,x) = \emptyset\); (ii) if \(x \in \text{cl} \bigcup \{g(y,x) : y \in A\}\), then \(x \in \text{cl} A\). Consequently, every monotonically normal \(T_1\)-space is an \(m_3\)-space. Subspaces of \(m_3\)-spaces are \(m_3\)-spaces. Box products of Hausdorff \(m_3\)-spaces are \(m_3\)-spaces. Countable \(m_3\)-spaces are \(M_1\)-spaces.
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54D70
Mathematics Subject Classification ID: 54E20
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 896579
zbMATH Keywords: monotonically normal space
zbMATH Keywords: stratifiable space
zbMATH Keywords: box product
zbMATH Keywords: \(m_ 1\)-space
zbMATH Keywords: \(m_ 3\)-space
zbMATH Keywords: \(m_ 2\)-space
zbMATH Keywords: strongly monotonically \(T_ 2\)-space
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
full work available at URL: 0166-8641(95)00069-0
OpenAlex ID: W2027050515
cites work: Q4064685
cites work: On stratifiable spaces
cites work: A Study of Monotonically Normal Spaces
cites work: Some generalizations of metric spaces
cites work: Q3047883
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cites work: Q3914054
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cites work: Monotonically Normal Spaces
cites work: \(M_ 3\)-spaces whose every point has a closure preserving outer base are \(M_ 1\)
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description: scientific article; zbMATH DE number 896579