Some topological properties of spaces between the Sorgenfrey and usual topologies on real number

TL;DR

This paper investigates various topological properties of a class of spaces on real numbers that blend Sorgenfrey and usual topologies, revealing conditions for zero-dimensionality, local compactness, and metrizability.

Contribution

It characterizes the topological properties of $H$-spaces on real numbers with mixed Sorgenfrey and usual topologies, including conditions for zero-dimensionality, $\sigma$-compactness, and metrizability.

Findings

Zero-dimensional iff $ e ackslash A$ is dense.

Locally compact iff $k_ ext{omega}$-space.

$\sigma$-compact implies $ e ackslash A$ is countable and nowhere dense.

Abstract

The $H$-space, denoted as $(\mathbb{R}, \tau_{A})$, has $\mathbb{R}$ as its point set and a basis consisting of usual open interval neighborhood at points of $A$ while taking Sorgenfrey neighborhoods at points of $\mathbb{R}$-$A$. In this paper, we mainly discuss some topological properties of $H$-spaces. In particular, we prove that, for any subset $A\subset \mathbb{R}$, (1) $(\mathbb{R}, \tau_{A})$ is zero-dimensional iff $\mathbb{R}\setminus A$ is dense in $(\mathbb{R}, \tau_{E})$; (2) $(\mathbb{R}, \tau_{A})$ is locally compact iff $(\mathbb{R}, \tau_{A})$ is a $k_{\omega}$-space; (3) if $(\mathbb{R}, \tau_{A})$ is $\sigma$-compact, then $\mathbb{R}\setminus A$ is countable and nowhere dense; if $\mathbb{R}\setminus A$ is countable and scattered, then $(\mathbb{R}, \tau_{A})$ is $\sigma$-compact; (4) $(\mathbb{R}, \tau_{A})^{\aleph_{0}}$ is perfectly subparacompact; (5) there exists a subset $A\subset\mathbb{R}$ such that $(\mathbb{R}, \tau_{A})$ is not quasi-metrizable; (6) $(\mathbb{R}, \tau_{A})$ is metrizable if and only if $(\mathbb{R}, \tau_{A})$ is a $\beta$-space.