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September 10, 2025Open Access

Several topological properties of H-spaces

Key Points

The space (R, ?A) is zero-dimensional if R is dense in (R, ?E), highlighting a critical topological link.
Locally compact status of (R, ?A) is equivalent to it being a k-space, emphasizing the depth of compactness in topology.
The space is ompact if R is countable and nowhere dense, which points to significant implications for h-space characteristics.
Metrizability is directly tied to whether (R, ?A) is a -space, showcasing essential relationships in topological structures.

Abstract

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.

Several topological properties of H-spaces

Key Points

Abstract

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