Some topological properties of Charming spaces

TL;DR

This paper explores the properties of charming spaces, showing that their squares may not be charming, providing characterizations, and proving that the Suslin number of certain charming rectifiable spaces is countable.

Contribution

It introduces new properties and characterizations of charming spaces and demonstrates that their squares can lack charm, advancing understanding of their topological structure.

Findings

Existence of charming space whose square is not charming

Characterizations of certain charming spaces

Suslin number of charming rectifiable spaces is countable

Abstract

In this paper, we mainly discuss the class of charming spaces, which was introduced by A.V. Arhangel'skii in [Remainders of metrizable spaces and a generalization of Lindel\"of $\Sigma$-spaces, Fund. Math., 215(2011), 87-100]. First, we show that there exists a charming space $X$ such that $X^{2}$ is not a charming space. Then we discuss some properties of charming spaces and give some characterizations of some class of charming spaces. Finally, we show that the Suslin number of an arbitrary charming rectifiable space $G$ is countable.