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On Extensions of Topologies

Published online by Cambridge University Press:  20 November 2018

Carlos J. R. Borges
Carlos J. R. Borges*
Affiliation:
University of California, Davis
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If (X, τ) is a topological space (with topology τ) and A is a subset of X, then the topology τ(A) = {U ⋃ (V ⋂ A)|U, V ∈ τ} is said to be a simple extension of τ. It seems that N. Levine introduced this concept in (4) and he proved, among other results, the following:

(A) If (X, τ) is a regular (completely regular) space and A is a closed subset of X, then (X, τ(A)) is a regular (completely regular) space.

(B) Let (X, τ) be a normal space, and A a closed subset of X. Then (X, τ(A)) is normal if and only if X — A is a normal subspace of (X, τ).

(C) Let (X, τ) be a countably compact (compact or Lindelöf) and A ∉ τ.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 474 - 487
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Borges, C. J. R., On stratifiable spaces, Pacific J. Math., 17 (1966), 1–16.Google Scholar
2. Céder, J. G., Some generalizations of metric spaces, Pacific J. Math., 11 (1961), 105–125.Google Scholar
3. Kelley, J. L., General topology (New York, 1955).Google Scholar
4. Levine, N., Simple extensions of topologies, Amer. Math. Monthly, 71 (1964), 22–25.Google Scholar