Maximal and minimal topologies

Cameron, Douglas E.

doi:10.1090/S0002-9947-1971-0281142-7

Maximal and minimal topologies
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by Douglas E. Cameron

Trans. Amer. Math. Soc. 160 (1971), 229-248

DOI: https://doi.org/10.1090/S0002-9947-1971-0281142-7

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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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