Maximal and minimal topologies

Abstract

A topological space ( X , T ) (X,T) with property R {\text {R}} is maximal R {\text {R}} (minimal R {\text {R}} ) if T T is a maximal (minimal) element in the set R ( X ) {\text {R}}(X) of all topologies on the set X X having property R {\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 P , Hausdorff quasi- P P , and P P topologies.