Spaces with unique Hausdorff extensions

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 extension. 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 Porterʹs 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 n 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 θ-equivalence relations defined by J.R. Porter and C. Votaw are equivalent.