-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

DOI:
https://doi.org/10.1090/S0002-9939-04-07336-8

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 -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 point-countable base, then is a -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:
https://doi.org/10.1090/S0002-9939-04-07336-8

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