I have a question to the relation between the usual notion of neighbourhood local base and open local base (which defines a local base via open sets instead of neighbourhoods).
In particular, I have a question to the answer given in the following thread :
what is the definition of local base
In the answer it is being said that "If we have any open set $U$ containing $x$, we have $x \in N \subseteq U$ for some $N \in \mathcal{N}$", where $\mathcal{N}$ is a neighbourhood local base.
But I am wondering why this statement should be true: If we take as our open set $U$ for instance an open set that is contained in an element $N \in \mathcal{N}$ of the local base, i.e. $U\subseteq N$, then there will not be a neighbourhood from the neighbourhood local base that is contained in said open set $U$, no? Since the elements from the neighbourhood local base are somewhat "the smallest" neighbourhoods that contain the point $x$.