Some topological properties of Charming spaces

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 $\mathrm{\Sigma }$-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.