Extending the discrete finite chain condition
type: Item
instance of: scholarly article
title: Extending the discrete finite chain condition
zbMATH Open document ID: 0793.54018
DOI: 10.1016/0166-8641(93)90107-O
author: Ian J. Tree
published in: Topology and its Applications
publication date: 1994-08-22T00:00:00Z
review text: Recall that a space \(X\) is feebly compact if every locally finite family of open subsets of \(X\) is finite; has the discrete finite chain condition (is DFCC) if every discrete collection of nonempty open sets of \(X\) is finite; and is pseudocompact if every continuous real valued function defined on \(X\) is bounded. It is known that in a regular space the first two of these properties are equivalent, and in a completely regular space all of them are equivalent. A space is called locally feebly compact (DFCC, pseudocompact) if each of its points has a feebly compact (DFCC, pseudocompact) neighborhood. In [Pac. J. Math. 36, 819-825 (1971; Zbl 0216.192)] the reviewer proved that every locally feebly compact, first countable zero-dimensional \(T_ 1\)-space has a feebly compact first countable zero-dimensional \(T_ 1\)- extension space. In [Fundam. Math. 91, 189-202 (1976; Zbl 0341.54016)] \textit{G. M. Reed} proved that every locally compact, separable Moore space can be embedded in a locally compact pseudocompact Moore space, and he asked whether every locally compact separable Moore space can be densely embedded. \textit{T. Terada} and \textit{J. Terasawa} proved in [Proc. Am. Math. Soc. 85, 95-99 (1982; Zbl 0477.54012)] that for Tykhonov spaces, every first countable locally pseudocompact (locally compact) space \(Y\) has a first countable (locally compact) pseudocompactification which is locally compact if \(Y\) is. Then in [Topology Appl. 27, No. 1, 11-28 (1987; Zbl 0633.54012)] the reviewer proved the following: Let \(Y\) be a first countable \(T_ 3\)-space in which the set \(B = \{y \in Y : Y\) is not locally feebly compact at \(y\}\) is countable, and there is a dense subset \(D\) of \(Y\) such that each point of \(D\) has a neighborhood base consisting of feebly compact open subsets of \(Y\). Then \(Y\) has a first countable feebly compact \(T_ 3\)-extension space \(X\). In the latter article the reviewer asked (1) if every locally feebly compact, first co…
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54E30
zbMATH DE Number: 474696
zbMATH Keywords: feebly compact space
zbMATH Keywords: dense embedding
zbMATH Keywords: discrete finite chain condition
zbMATH Keywords: DFCC
zbMATH Keywords: Moore space
zbMATH Keywords: pseudocompactification
reviewed by: Robert M. Stephenson Jr.
MaRDI profile type: Publication
cites work: On Properties Characterizing Pseudo-Compact Spaces
cites work: First countable pseudocompactifications
cites work: Stone-Čech compactification of a product
cites work: Q3706143
cites work: On chain conditions in moore spaces
cites work: On subspaces of separable first countable $T_2$-spaces
cites work: Moore-closed and first countable feebly compact extension spaces
cites work: Maximal Extensions of First-Countable Spaces
cites work: Q5605131
description: scientific article; zbMATH DE number 474696