On minimal Hausdorff first countable spaces
type: Item
instance of: scholarly article
title: On minimal Hausdorff first countable spaces
zbMATH Open document ID: 0539.54011
author: T. G. Raghavan
author: I. L. Reilly
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1983-01-01T00:00:00Z
review text: A topological space X is called an \(E_ 1\) space if for every point x of X there is a countable collection \({\mathcal B}\) of closed neighborhoods of x such that \(\cap {\mathcal B}=\{x\}.\) By an \(\alpha\)-minimal \(E_ 1\) space the authors mean an \(E_ 1\) space X such that: for every closed subset A of X and countable cover \({\mathcal V}\) of A by open subsets of X, there is a finite subset \({\mathcal F}\) of \({\mathcal V}\) with \(A\subset cl(\cup {\mathcal F}).\) By a \(\beta\)-minimal \(E_ 1\) space the authors mean an \(E_ 1\) space X which has the following property: whenever \({\mathcal F}\) is a countable open filter base on X such that \(\cap {\mathcal F}=\cap \{cl(F):\quad F\in {\mathcal F}\},\) then \({\mathcal F}\) is a base for the family of all neighborhoods of the set \(\cap {\mathcal F}\). The authors study \(\alpha\)-minimal and \(\beta\)- minimal \(E_ 1\) spaces and obtain a number of interesting results concerning them, such as the following: An \(E_ 1\) space X is \(\beta\)- minimal, \(E_ 1\) iff every continuous mapping from X onto an \(E_ 1\) space is a closed mapping. Several examples are given.
Mathematics Subject Classification ID: 54D25
Mathematics Subject Classification ID: 54D20
zbMATH DE Number: 3857943
zbMATH Keywords: \(\alpha\)-minimal \(E_ 1\) space
reviewed by: Robert M. Stephenson Jr.
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3857943