This should be easy, but apparantly not for me. Let G be a topological group, and let $\mathcal{N}$ be a neighbourhood base for the identity element $e$ of $G$. Then for all $N_1,N_2 \in \mathcal{N}$, there exists an $N' \in \mathcal{N}$ such that $e \in N'\subset N_1 \cap N_2.$
This means for example that every $N \in \mathcal{N}$ contains the identity element, which seems strange to me.