$d(X) \le w(X)$ using the axiom of choice

Question

$d(X) = \min ({ \vert D\vert : \overline{D} = X})$

$w(X) = \min (\vert B\vert : B \textrm{ is a base for }X) $

WTS: $d(X) \le w(X)$

I know that if I have a set $B : B = w(X)$ , I can apply the axiom of choice to create a dense subset. But how does doing this make my subset dense?

Answer 1

Take a point $b$ for each element of $B$ (the elements of $B$ are subsets of $X$). Now, any open set of $X$ is a union of elements of $B$...