1 ) Why $X = \{ 0 \} \cup \{ 1 / n : n \in \mathbb{N} \}$ is $0$-dimensional ?
2 ) Let $X$ be a space and $G$ a topological group, Why If $X$ is $0$-dimensional in the sense of ind, then $X$ is $G^{**}$-regular ?
Note 1 : $X$ is $0$-dimensional if has a base of clopen sets.
Note 2 : $G^{⋆⋆}$-regular provided that, whenever $F$ is a closed subset of $X$, $x \in X \setminus F$ and $g \in G$, there exists $f \in Cp(X,G)$ such that $f(x) = g$ and $f(F) \subseteq \{e\}$.