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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$D$-spaces and finite unions


Author: Alexander Arhangel'skii
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2163-2170
MSC (2000): Primary 54D20; Secondary 54F99
Published electronically: February 9, 2004
MathSciNet review: 2053991
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Abstract: This article is a continuation of a recent paper by the author and R. Z. Buzyakova. New results are obtained in the direction of the next natural question: how complex can a space be that is the union of two (of a finite family) ``nice" subspaces? Our approach is based on the notion of a $D$-space introduced by E. van Douwen and on a generalization of this notion, the notion of $aD$-space. It is proved that if a space $X$ is the union of a finite family of subparacompact subspaces, then $X$ is an $aD$-space. Under $(CH)$, it follows that if a separable normal $T_1$-space $X$ is the union of a finite number of subparacompact subspaces, then $X$ is Lindelöf. It is also established that if a regular space $X$ is the union of a finite family of subspaces with a point-countable base, then $X$ is a $D$-space. Finally, a certain structure theorem for unions of finite families of spaces with a point-countable base is established, and numerous corollaries are derived from it. Also, many new open problems are formulated.


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Additional Information

Alexander Arhangel'skii
Affiliation: Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701
Email: arhangel@math.ohiou.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07336-8
PII: S 0002-9939(04)07336-8
Keywords: $D$-space, point-countable base, extent, subparacompact space, Lindel\"of degree, $aD$-space
Received by editor(s): October 21, 2002
Received by editor(s) in revised form: April 14, 2003
Published electronically: February 9, 2004
Communicated by: Alan Dow
Article copyright: © Copyright 2004 American Mathematical Society