spaces and finite unions
Author:
Alexander Arhangel'skii
Translated by:
Journal:
Proc. Amer. Math. Soc. 132 (2004), 21632170
MSC (2000):
Primary 54D20; Secondary 54F99
Published electronically:
February 9, 2004
MathSciNet review:
2053991
Fulltext PDF Free Access
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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 space introduced by E. van Douwen and on a generalization of this notion, the notion of space. It is proved that if a space is the union of a finite family of subparacompact subspaces, then is an space. Under , it follows that if a separable normal space is the union of a finite number of subparacompact subspaces, then is Lindelöf. It is also established that if a regular space is the union of a finite family of subspaces with a pointcountable base, then is a space. Finally, a certain structure theorem for unions of finite families of spaces with a pointcountable 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/S0002993904073368
PII:
S 00029939(04)073368
Keywords:
$D$space,
pointcountable 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
