Minimal Hausdorff and maximal compact spaces

Abstract

Given two topologies J 1 , J 2 on a set X , J 1 is said to be coarser than J 2 , written J 1 ≦ J 2 , if every set open under J 1 is open under J 2 . A minimal Hausdorff space is then one for which there is no coarser Hausdorff topology etc. Vaidyanathaswamy [4] showed that every compact Hausdorff space is both maximal compact and minimal Hausdorff. This raised the question of whether there exist minimal Hausdorff non-compact spaces and/or maximal compact non-Hausdorff spaces. These questions were in fact answered in the affirmative by Ramanathan [2], Balachandran [1], and Hing Tong [3]. Their examples were, however, all on countable sets, and the topology constructed to answer one question bore no relation to the topology answering the second. In particular, the minimal Hausdorff non-compact topologies were not finer than any maximal compact topology.