I have a couple of questions:
I need a hint to show that if X is a regular, Hausdorff and separable space, then there exists a basis $B$ of the topology of X such that $|B|\le 2^{\aleph_0}$.
How do you show that $[0,1]^{A}$ is separable if and only if $|A|\le 2^{\aleph_0}$
Thank you.
(1) First show that since $X$ is regular and Hausdorff, it has a base of regular open sets, i.e., open sets $U$ such that $\operatorname{int}\operatorname{cl}U=U$. Let $\mathscr{B}=\{U\subseteq X:\operatorname{int}\operatorname{cl}U=U\}$, and let $D$ be a countable dense subset of $X$. Show that if $U\in\mathscr{B}$, then $U=\operatorname{int}\operatorname{cl}(U\cap D)$. Use this to show that there is an injection (one-to-one function) from $\mathscr{B}$ into $\wp(D)$, and conclude that $|\mathscr{B}|\le|\wp(D)|$.
(2) One direction of this is a special case of the Hewitt-Marczewski-Pondiczery theorem; Hewitt’s original paper can be found here. This result itself is Theorem $16.4$c in Willard’s General Topology; it is also proved here at Ask a Topologist by Henno Brandsma.
