On the role of finite, hereditarily normal spaces and maps in the genesis of compact Hausdorff spaces.
type: Item
instance of: scholarly article
title: On the role of finite, hereditarily normal spaces and maps in the genesis of compact Hausdorff spaces.
zbMATH Open document ID: 1043.54005
DOI: 10.1016/S0166-8641(03)00181-0
author: Ralph Kopperman
author: Richard G. Wilson
published in: Topology and its Applications
publication date: 2004-02-03T00:00:00Z
review text: The problems to which the reviewed paper is devoted originated from a celebrated theorem by \textit{A. Kurosch} [Compos. Math. 2, 471--476 (1935; Zbl 0012.32002)], according to which a compact Hausdorff space can be obtained as an inverse limit of finite \(T_0\)-spaces with continuous bonding maps. The method was developed by \textit{P. S. Alexandroff} and was applied by \textit{J. Flachsmeyer} [Math. Ann. 144, 253--274 (1961; Zbl 0118.17701)] in the theory of Hausdorff compactifications. The authors of the paper under review distinguish some special classes of finite \(T_0\)-spaces and some special kinds of bonding maps to obtain new special results in the area described above. The authors regard as one of the main results the Theorem 3.15 according to which the Hausdorff reflection of the inverse limit of normal finite \(T_0\)-spaces with continuous bonding maps is hereditarily disconnected. From the proof of this theorem the authors obtain as a corollary that the inverse limit of finite connected normal \(T_0\)-spaces with continuous bonding maps is normal and connected.
reviewed by: Jerzy Mioduszewski
Mathematics Subject Classification ID: 54B35
zbMATH DE Number: 2037242
zbMATH Keywords: finite \(T_0\)-spaces
zbMATH Keywords: inverse limit
zbMATH Keywords: normal
zbMATH Keywords: connected
MaRDI profile type: Publication
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cites work: Q3270120
cites work: Joincompact spaces, continuous lattices and \(C^*\)-algebras
cites work: Prime Ideal Structure in Commutative Rings
cites work: The chainable continua are the spaces approximated by finite COTS
cites work: Q4500944
cites work: The Approximation of Compacta by Finite T0-Spaces
cites work: Asymmetry and duality in topology
cites work: Compact compatible topologies for posets and graphs
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description: scientific article; zbMATH DE number 2037242