Results on spaces between the Sorgenfrey and usual topologies on R

In [4], Hattori described a natural set of spaces, H-spaces, between the usual topology of the real numbers R and that of the Sorgenfrey line S, and initiated a study of their properties. Hattori and Chatyrko developed a considerable understanding of H-spaces in [1] and [2]. The H-space based on A, denoted (R,τA), has R as its point set and a basis consisting of usual R neighborhoods at points of A while taking Sorgenfrey neighborhoods at points of R−A. In this paper, we address some of the open questions found in these papers and also describe some H-spaces with unusual properties. In particular, we show that an H-space is topologically complete if and only if R−A is a countable set and that any continuous bijective function h:(R,τA)→(R,τB) must have only countably many points of A which are mapped to R−B. We also categorize the subsets A of R for which the H-space (R,τA) is homeomorphic to S as exactly the scattered subsets of R. Furthermore, we exhibit a collection of 2c distinct reversible H-spaces, no two of which are homeomorphic. In fact there exist no continuous bijections between any two of them, and the only continuous auto-bijection on any one of them is the identity.