A connected pseudocompact space
type: Item
instance of: scholarly article
title: A connected pseudocompact space
zbMATH Open document ID: 0805.54034
DOI: 10.1016/0166-8641(94)90047-7
published in: Topology and its Applications
publication date: 1995-02-05T00:00:00Z
review text: In this article a space \(X\) is called pseudocompact if every discrete collection of open subsets of \(X\) is finite. Recall that a set \(A\) is said to be conditionally compact or relatively countably compact in a space \(X\) if every infinite subset of \(A\) has a limit point in \(X\). At the 1990 Summer Conference in General Topology at Long Island University the reviewer asked if there is a connected, pseudocompact, first countable, regular space which is not countably compact. The author obtains the very nice result: (CH) There is a first countable, separable, connected, pseudocompact, Tikhonov space without a dense relatively countably compact subset. The construction is a very complicated and clever blending of an Ostaszewski-type construction with a \(\Psi\)-space type of construction. The space produced actually contains \(\mathbb{R}\) as a dense open subspace. As was the case with the article by \textit{Ian J. Tree} [Topology Appl. 52, No. 3, 267-278 (1993; Zbl 0793.54018)], a most worthwhile accomplishment of the author is the generality achieved by devising a method of pseudocompactifying a given space \(X\) that involves adding to \(X\) something more than a disjoint closed discrete set. The author also obtains some results on pseudocompactifications, such as: Every locally pseudocompact first countable regular space with a dense locally conditionally compact subset can be embedded in a pseudocompact first countable regular space. Moreover, he introduces a property SLP which is implied by either locally countably compact or the possession of a dense conditionally compact subset, and yet implies locally pseudocompact such that every SLP, first countable, regular space can be embedded in a pseudocompact, first countable, regular space. (A space \(X\) is called strongly locally pseudocompact or SLP if there is a dense \(D\subset X\) such that, for all \(x\in X\), there is a closed neighborhood \(U\) of \(x\) such that \(D\cap U\) is relatively countably c…
Mathematics Subject Classification ID: 54G20
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54D35
zbMATH DE Number: 600035
zbMATH Keywords: Mrówka-Isbell space
zbMATH Keywords: Ostaszewski space
zbMATH Keywords: connected, pseudocompact, first countable, regular space
zbMATH Keywords: pseudocompactifications
author: Stephen Watson
reviewed by: Robert M. Stephenson Jr.
MaRDI profile type: Publication
full work available at URL: 0166-8641(94)90047-7
OpenAlex ID: W2046626873
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description: scientific article; zbMATH DE number 600035