Suppose that $X$ has countable basis, then every open covering of $X$ contains a countable subcollection covering $X$.
Proof
{$U_i;i\in I$} is open cover for ($X,\tau$) and $B$ = {$V_n;n\in N$} is countable base for topology $\tau$.We have that $X=\cup_i U_i$.For every point $x$ in $X$ let's choose any $U_i$ s.t. $x \in U_i$.Denote that $U_i$ as $U_x$.There exists $V_x \in B$ s.t. $x\in V_x\subset U_x$.Collection of $V_x$ is countable.For every $V_x$ choose $U_x$ s.t. $V_x\subset U_x$.Because $X=\cup_xV_x$ follows that $X=\cup U_x$ and number of $U_x$-s are countable.
I understood proof only last part I can't understand.Why we need that number of $U_x$ to be countable?
We have already open cover for $X$ and found {$V_x$} which is countable subcollection covering $X$.
Best Answer
{$U_i;i\in I$} is open cover for ($X,\tau$) and $B$ = {$V_n;n\in N$} is countable base for topology $\tau$.We have that $X=\cup_i U_i$.For every point $x$ in $X$ let's choose any $U_i$ s.t. $x \in U_i$.Denote that $U_i$ as $U_x$.There exists $V_x \in B$ s.t. $x\in V_x\subset U_x$.
Now as these $\{V_{x}\}$ which correspond to the $U_{x}$'s is a subcollection of $B$. Hence is countable as a subset of countable set is countable. Now it's just a matter of indexing. Explicitly we were looking at $U_{0}=\{U_{x}\in \{U_{i}\}_{i\in I} : x\in V_{x}\subset U_{x}\}$. correspondingly we have $B_{0}=\{V_{x}: V_{x}\subset U_{x}\}$. These sets $B_{0}$ and $U_{0}$ are in bijection and $B_{0}$ being a subset of countable set is countable.
Write out all such $V_{x}$'s as a sequence say $\{V_{1},V_{2},.....\}$. Correspondingly we have $\{U_{1},U_{2},....\}$.
Thus $\{U_{k}\}_{k=1}^{\infty}$ is a countable subcover of $\{U_{i}\}_{i\in I}$