Which spaces have a coarser connected Hausdorff topology?
type: Item
instance of: scholarly article
title: Which spaces have a coarser connected Hausdorff topology?
zbMATH Open document ID: 1012.54021
DOI: 10.1016/S0166-8641(01)00274-7
author: William Fleissner
author: J. R. Porter
author: Judith Roitman
published in: Topology and its Applications
publication date: 2003-03-10T00:00:00Z
review text: Some answers to the question in the title are considered; that is to say, the authors seek necessary and sufficient conditions in order that there exists a continuous bijection (or condensation) from a space \((X,\tau)\) to a connected Hausdorff space. Section 2 of the paper is dedicated to obtaining some necessary conditions for the existence of a weaker connected topology while in Sections 4 and 5 some sufficient conditions are given using the somewhat technical concepts of \(\kappa\)-epoxicity and \(\kappa\)-super-epoxicity. As corollaries to the technical results, it is shown that every cardinal has a weaker connected Hausdorff topology (for the cardinal \(\omega_1\) this was first shown by \textit{S. Christodoulou} [Quest. Answers Gen. Topology 10, No. 2, 143-148 (1992; Zbl 0782.54002)]) and that \((2^{\omega})^{+}+\omega\) is the smallest non-compact ordinal with no such weaker topology. Section 6 considers extensions of the discrete space \(\omega\) and the main theorem of this section states that a locally countable, locally compact extension of \(\omega\) of cardinality \(\omega_1\) has a weaker connected Hausdorff topology if \(\omega_1< {\mathfrak p}\). The final section considers spaces between \(\omega\) and \(\beta\omega\) and includes an interesting example of a Hausdorff space which can be condensed onto a connected Hausdorff space but whose semiregularization cannot be so condensed.
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D05
zbMATH DE Number: 1879121
zbMATH Keywords: weaker connected topology
zbMATH Keywords: condensation
zbMATH Keywords: Franklin-Rajagopalan space
zbMATH Keywords: extension
zbMATH Keywords: minimal Hausdorff space
reviewed by: Richard G. Wilson
MaRDI profile type: Publication
cites work: A connected countable Hausdorff space
cites work: Q5186348
cites work: Q3685639
cites work: Q3994935
cites work: Zur Theorie der H-abgeschlossenen Erweiterungen
cites work: Some Examples in Topology
cites work: A note on compactifications
cites work: Q4828842
cites work: Q4039810
cites work: Q4383370
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description: scientific article; zbMATH DE number 1879121