Normally supercompact spaces and completely distributive posets☆☆This work was supported by the National Education Committee of China for outstanding youths and by the National Education Committee of China for scholars returning from abroad. This paper was completed when the author was visiting Osaka Kyoiku University.

Abstract

A Hausdorff space X is called normally supercompact (NS for short) if it has a subbase S such that (1) every cover consisting of elements of S has a subcover consisting of at most two elements, and (2) for any pair A,B of elements of S if A∪B=X then there exist C,D∈ S such that C∩D=∅ and A∪C=B∪D=X . A poset L is called a completely distributive poset (CDP for short) if (3) every nonempty subset has an inf, (4) every subset in which every pair has an upper bound has a sup, and (5) the distributive law holds for any existing sups and existing infs. In this paper, we prove that the category of all NS spaces and the category of all CDP's are isomorphic. As a result we deduce that the order in a connected compact linearly ordered space is unique. Moreover, we also get a corresponding result for zero-dimensional NS spaces. In particular, we show that a space is zero-dimensional NS if and only if it has a subbase consisting of clopen sets satisfying (1) and (2).