Let \(X\) be a first countable space . Every point \(p\in X\) has a nested neigbourhood basis .
Proof
Let \(p\in X\). Since \(X\) is first countable there is a countable neighbourhood basis
\((U_i)\). Then \(V_i=U_1\cap \cdots \cap U_i\) is a nested neighbourhood basis (note ever \(V_i\) is open, since finite intersections of open sets are open).