A $T_B$ space which is not Katetov $T_B$

Abstract

In 1943, E. Hewitt [1] proved the beautiful theorem that a compact Hausdorff space is minimal Hausdorff and maximal compact. Restating this result in more detail, if (X, is a compact Hausdorff space and (X, r) and (X, z) are spaces z' £ z £ z, then (X, z') is not Hausdorff, and (X, z) is not compact. The converses to this theorem are appealing but false. There are noncompact minmal Hausdorff spaces [2] and non Hausdorff maximal compact spaces [2]. A compact space is maximal compact if every compact set is closed [3]. Let us call spaces in which all compact sets are closed TB spaces, as this notion can be thought of as a separation axiom between 7 and T2. They are also called KC spaces. R. Larson [4] asked whether a space is maximal compact iff it is minimal TB. A related question is whether every TB topology is Katetov TB, that is whether every TB topology contains a minimal TB topology. The author wishes to thank Douglas Cameron for bringing these questions to his attention. In this paper we construct a TB not Katetov TB tpace. The point set of all spaces in this paper will be the countable ordinals. To avoid ambiguity, we will refer to the first uncountable ordinal (and cardinal) as o>l5 and to the point set of the spaces as Q. A typical point of Q will be xa, where a a) will be called S(a), the successors of a. The usual topology on O, generated by {P(a): a i) U {S(a) : a < coi} will be called «. The cardinality of a set S will be denoted \S\.