A note on maximally resolvable spaces
type: Item
instance of: scholarly article
title: A note on maximally resolvable spaces
zbMATH Open document ID: 0705.54004
DOI: 10.1155/S0161171290000746
author: V. Tzannes
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46504
review text: Summary: \textit{A. G. El'kin} [Vestn. Mosk. Univ. Ser. I 24, No.4, 66-70 (1969; Zbl 0183.512)] poses the question as to whether any uncountable cardinal number can be the dispersion character of a Hausdorff maximally resolvable space. In this note we prove that every cardinal number \(\aleph \geq \aleph_ 1\) can be the dispersion character of a metric (hence, maximally resolvable) connected, locally connected space. We also prove that every cardinal number \(\aleph \geq \aleph_ 0\) can be the dispersion character of a Hausdorff (resp. Urysohn, almost regular) maximally resolvable space X with the following properties: 1) Every continuous real-valued function of X is constant, 2) For every point x of X, every open neighborhood U of x, contains an open neighborhood V of x such that every continuous real-valued function of V is constant. Hence the space X is connected and locally connected and therefore there exists a countable connected locally connected Hausdorff (resp. Urysohn or almost regular) maximally resolvable space (not satisfying the first axiom of countability).
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54G20
zbMATH DE Number: 4157449
zbMATH Keywords: dispersion character
zbMATH Keywords: Hausdorff maximally resolvable space
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4157449