Suppose that you were given a sequence $M_1, M_2, M_3, \ldots$ of neighborhoods of a point $p$ in a space $(X, \tau)$ such that for every neighborhood $N$ of $p, \ \exists n \in \mathbb N$ such that $M_n \subseteq N.$ How could you modify the given material to produce a countable local base at $p$?
Would the following work? Or am I misunderstanding the problem?
For every $N_i$ of $p$, there’s $n \in \mathbb N$ s.t. $n = i+ 1$ and $M_n \subseteq N_i$?
Or
for every $N$ of $p, \ \exists n \in \mathbb N$ such that $M_n \subseteq N \subseteq M_{n - 1}?$