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Two R-Closed Spaces

Published online by Cambridge University Press:  20 November 2018

R. M. Stephenson
R. M. Stephenson*
Affiliation:
The University of North Carolina, Chapel Hill, North Carolina
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Throughout this paper all hypothesized spaces are T1. A regular space is called R-closed[11](regular-closed [7] or, equivalently, regular-complete [2]) provided that it is a closed subset of any regular space in which it can be embedded. A regular space (X, ℐ) is called minimal regular [2; 4] if there exists no regular topology on X which is strictly weaker than J. We shall call a regular space X strongly minimal regular provided that each point x ∈ X has a fundamental system of neighbourhoods such that for every V ∈ , X\V is an R-closed space.

In §2 we note that a strongly minimal regular space is minimal regular, but we do not know if the converse holds. M. P. Berri and R. H. Sorgenfrey [4] proved that a minimal regular space is R-closed, and Horst Herrlich [7] gave an example of an R-closed space that is not minimal regular.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 2 , April 1972 , pp. 286 - 292
Copyright
Copyright © Canadian Mathematical Society 1972

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