neighborhood basis on a set Given a topological space $(X,\tau)$ I know perfectly what is a neighborhood basis at $x\in X$.
Now suppose that $X$ is just a set, fix a point $x\in X$ and consider a collection $\mathcal B(x)$ of subsets of $X$ containing $x$. What are the properties that I have to check for $B(x)$ in order to ensure that there exists a topology on $X$ such that a local basis at $x$ is $B(x)$?
This question arises from the following consideration. Very often topologies in books are presented in a weird way; the author simply says "Consider on the set $X$ the topology $\tau$ having this particular neighborhood basis". I'd like to check that what he claims to be a "local basis" is indeed a good choice.

Assuming that you have a family $\mathscr{B}(x)$ for each $x\in X$, the key thing to check is that if $B_x\in\mathscr{B}(x)$, $B_y\in\mathscr{B}(y)$, and $z\in B_x\cap B_y$, then there is a $B\in\mathscr{B}(z)$ such that $B\subseteq B_x\cap B_y$. This ensures that $\bigcup_{x\in X}\mathscr{B}(x)$ is a base for a topology on $X$.
