Several topological properties of H-spaces

AnH-space, denoted as (R, ?A), hasRas its point set and a basis consisting of usual open interval neighborhoods at points of A while taking Sorgenfrey neighborhoods at points of R \ A. In this paper, we mainly discuss some topological properties of H-spaces. In particular, we prove that, for any subset A ? R, (1) (R, ?A) is zero-dimensional iff R \ A is dense in (R, ?E), where ?E is the natural topology on R; (2) (R, ?A) is locally compact iff (R, ?A) is a k?-space; (3) if (R, ?A) is ?-compact, then R is countable and nowhere dense; if R is countable and scattered in the real line, then (R, ?A) is ?-compact; (4)Q?i=1(R, ?Ai) is perfectly subparacompact, where each Ai is a subset of R; (5) there exists a subset A ? R such that (R, ?A) is not quasi-metrizable; (6) (R, ?A) is metrizable if and only if (R, ?A) is a ?-space.