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On Unions of Metrizable Subspaces

Published online by Cambridge University Press:  20 November 2018

E. K. Van Douwen ,
D. J. Lutzer ,
J. Pelant  and
G. M. Reed
E. K. Van Douwen
Affiliation:
Ohio University, Athens, Ohio
D. J. Lutzer
Affiliation:
Texas Tech University, Lubbock, Texas
J. Pelant
Affiliation:
Institute of Mathematics, ČSAV Prague, Czechoslovakia
G. M. Reed
Affiliation:
Institute of Mathematics, ČSAV Prague, Czechoslovakia
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In this paper we study the question “which generalized metric spaces can be written as the union of k (closed) metrizable subspaces, where k is a cardinal number with k ≤ c?“ Questions of this type first arose in [16] where J. Nagata asked for examples of certain generalized metric spaces which could not be written as the union of countably many closed metrizable subspaces. Using Baire Category arguments, Fitzpatrick provided the required examples in [12]. We begin this paper by sharpening Fitzpatrick's examples, showing in Section 2 that there is a Moore space which is not the union of countably many metrizable subspaces of any kind. Then in Section 3 we present a positive result, proving that any cr-space, and a fortiori any Moore space, can be written as the union of c = 2ω0 closed metrizable subspaces.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 76 - 85
Copyright
Copyright © Canadian Mathematical Society 1980

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