Are Relatively compact space finite union of open subsets?

Question

Suppose that $X$ is a topological space and $U\subset X$ is an open set which is relatively compact, i.e. $\overline{U}$ is compact. If $\mathcal{B}$ is a basis of $X$ is it true that $U$ is finite union of elements of $\mathcal{B}$?

If the first question has negative answer what we can say about $X=\mathbb{R}^n$ and $\mathcal{B}$ the canonical open cover of balls of rational radius and center in $\mathbb{Q}^n$?

Answer 1

Take $X = \mathbb R^2$ and $U = (0,1)\times (0,1)$. Take $\mathcal B$ to be the canonical open cover of open balls - it doesn't matter if you restrict to ones with rational centre and radius or not. $\bar U$ is compact, yet $U$ is not a finite union of balls, since it's impossible to "fill in the corners" of $U$ with only finitely many balls.