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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Maximal and minimal topologies


Author: Douglas E. Cameron
Journal: Trans. Amer. Math. Soc. 160 (1971), 229-248
MSC: Primary 54.20
DOI: https://doi.org/10.1090/S0002-9947-1971-0281142-7
MathSciNet review: 0281142
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Abstract: A topological space $ (X,T)$ with property $ {\text{R}}$ is maximal $ {\text{R}}$ (minimal $ {\text{R}}$) if $ T$ is a maximal (minimal) element in the set $ {\text{R}}(X)$ of all topologies on the set $ X$ having property $ {\text{R}}$ with the partial ordering of set inclusions. The properties of maximal topologies for compactness, countable compactness, sequential compactness, Bolzano-Weierstrass compactness, and Lindelöf are investigated and the relations between these spaces are investigated. The question of whether any space having one of these properties has a strictly stronger maximal topology is investigated. Some interesting product theorems are discussed. The properties of minimal topologies and their relationships are discussed for the quasi-$ P$, Hausdorff quasi-$ P$, and $ P$ topologies.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0281142-7
Keywords: Maximal topologies, minimal topologies, compactness, countable compactness, sequential compactness, Bolzano-Weierstrass compactness, Lindelöf, quasi-$ P$-spaces, Hausdorff quasi-$ P$-spaces, $ P$-spaces, strongly $ {\text{R}}$ topologies
Article copyright: © Copyright 1971 American Mathematical Society