Spaces with unique Hausdorff extensions

Douglas D. Mooney
1995 Topology and its Applications  
H-closed extensions of Hausdorff spaces have been studied extensively as a generalization of compactifications of Tychonoff spaces. The collection of H-closed extensions of a space is known to have an upper semilattice structure. Little work has been done to characterize spaces whose collections of H-closed extensions have specified upper semilattice structures. In 1970 J.R. Porter found necessary and sufficient conditions on a space so that it would have exactly one one-point H-closed
more » ... . He asked for a characterization of those spaces which have exactly one H-closed extension. This is the same as having exactly one Hausdorff extension. In this paper we answer Porter's question and give an example of such a space. Topological sums of this space give spaces which have two, five, or in general, p(n) many H-closed extensions where p(n) is the number of ways a set of size it can be partitioned. This space is also an example of a space with exactly one free prime open filter which gives an answer to a question asked by J. Pelant, P. Simon, and J. Vaughan. As a preliminary for obtaining the above results, we find necessary and sufficient conditions on a space so that the S-and f&equivalence relations defined by J.R. Porter and C. Votaw are equivalent. Proposition 4.6 [18]. If Sr is a free open filter on X, then P(g) is a compact subset of aX\X. Conversely, if A is a compact subset of aX\X, then f-j A is a free open filter and P( n A) = A. (Note that n A is the intersection of open filters and is, therefore, an open filter .) Observe that if Y is an extension of X, p E Y\X, and fY is the KatEtov map, then P(Op) = f;(p). A consequence of Proposition 4.6 is the following very useful result. Proposition 4.7 [18]. Each H-closed extension determines a partition of uX\X into compact sets (in the topology of aX\X>. Conversely each partition of uX\X into compact sets determines an H-closed extension which is unique up to &equiualence. Proposition 4.7 implies that every partition of uX\X into compact sets determines a &equivalence class and conversely every &equivalence class determines a partition of uX\X into compact sets.
doi:10.1016/0166-8641(94)00031-w fatcat:zt73d7swdzandews5cmlv2zufu