I have two problems:
1.- Let $X$ be a compact Hausdorff space, then $X$ has a basis with cardinality less than or equal to $|X|$.
2.- Let $X$ be a Hausdorff space and $D$ a dense subset in $X$, then $|X|\leq|P(P(D))|$, where $P(D)$ is the power set of $D$.
If somebody know where I can find the proofs, tell me please.