Suppose $\langle X,\mathscr{O}\rangle$ is a topological space and let $\mathscr{O}_x$ be the family of all open neighbourhoods of $x\in X$. Let $\mathscr{F}$ be the filter generated from $\mathscr{O}_x$ (the neighbourhood filter at $x$): $$\mathscr{F}=\{Q\in2^X\mid(\exists V\in\mathscr{O}_x)\,V\subseteq Q\} $$

What I am interested in is under what conditions (if any) can we guarantee existence of a chain $C\subseteq\mathscr{O}_x$ (may be uncountable) such that the filter $\mathscr{F}_C$ generated by $C$ is equal to $\mathscr{F}$.