Maximal and Minimal Topologieso

A topological space (X, T) with property R is maximal R (minimal R) if T is a maximal (minimal) element in the set R(X) of all topologies on the set X having property 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-i7, Hausdorff quasi-P, and P topologies. I. Background and introduction. For a given topological property R and a set X, we let R(A') denote the set of all topologies on X which have property R and observe that R(A) is partially ordered by set inclusion. A topological space (X, T) is maximal R (R-maximal) provided that F is a maximal element in R(A'). A topological space (A, T) is minimal R (R-minimal) if F is a minimal element in R(A"). A topology T' on the set A'is finer than a topology 7" if F 3 T; the topology F is said to be coarser than the topology T'. The concept of minimal topologies was first introduced in 1939 by A. S. Parhomenko [19] when he showed that compact Hausdorff spaces are minimal HausdorfT. Four years later E. Hewitt [12] proved that compact Hausdorff spaces are maximal compact as well as being minimal Hausdorff. In 1947 R. Vaidyanathaswamy [33] asked if there existed noncompact minimal Hausdorff spaces or non-Hausdorff maximal compact spaces. The same year A. Ramanathan [22], [23] showed that such minimal Hausdorff spaces existed and characterized all minimal Hausdorff spaces. In 1948 Hing Tong [30] constructed an example of a maximal compact space which was not Hausdorff and thus answered the other part of Vaidyanathaswamy's question. In 1948 A. Ramanathan [24] proved that a topological space is maximal compact Received by the editors June 29, 1970 and, in revised form, November 18, 1970. AMS 1970 subject classifications. Primary 54A10, 54B05, 54B10, 54C20, 54D20, 54D25, 54D30, 54D35; Secondary 54B05, 54D60.