Countable collection of open sets in space with certain property

Question

Let $(X,\tau)$ be a topological space.

Let $\mathcal{F}=\{U_n\}_{n\geq 1}$ be a countable collection of open sets such that for any $O\in \tau$ $\exists N:$ $U_N\subset O$.

I want to show that $\mathcal{F}$ is basis for topology on $X$.

First of all we have to check that it is basis.

1) Take any $x\in X$ then $\exists N$: $x\in U_N$ because $X$ is itself open set.

2) But if $x\in U_k\cap U_m$ then since their intersection is also open set then $\exists N$: $U_N\subset U_k\cap U_m$. If $x\in U_N$ then it is ok, but if $x\notin U_N$?

Can anyone explain it to me please?

Answer 1

This $\mathcal{F}$ need not be a base for the topology (it's just a $\pi$-base by definition). For the Mrówka $\Psi$-space (or $\beta\Bbb N$) we have a countable dense set of isolated points (the singletons of that set can form $\mathcal{F}$) in a space with uncountable weight.

So what you want to show is impossible in general.