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Closed Maps and Paracompact Spaces

Published online by Cambridge University Press:  20 November 2018

H. L. Shapiro
H. L. Shapiro*
Affiliation:
The Pennsylvania State University, University Park, Pennsylvania
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Let ƒ be a map from a topological space X into a topological space F. We say that ƒ is proper in case ƒ is closed continuous and ƒ-1(y) is compact for all y ∊ Y. Proper maps have been extensively studied, see for example (3, Chapter I, §10) or (6). (The definition of a proper map given above is different from, but equivalent to, that given by Bourbaki in (3). In (6) only surjective proper maps are considered and these maps are called ƒitting maps.) It is known that if ƒ is a proper map, then X is compact if and only if ƒ(X) is compact, and X is paracompact if and only if ƒ(X) is paracompact.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 513 - 519
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

The results reported in this paper consist principally of a portion of the author's doctoral dissertation written at Purdue University under the direction of Professor Robert L. Blair, to whom the author wishes to express his appreciation. This research was supported in part by a grant from the National Science Foundation.

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