Given a first countable space it is always possible to build a neighborhood basis of a point for which the inclusion is a total order. Let's say $\lbrace V_n \rbrace _{n \in \mathbb{N}}$ is any countable neighborhood basis of that point, then define $\lbrace U_n \rbrace _{n \in \mathbb{N}}$ as follows:
$U_1=V_1$
$U_k=V_k \cap V_{k-1} \quad \mbox{if $k>1$}$
It's easy to check that $\lbrace U_n \rbrace _{n \in \mathbb{N}}$ equipped with the inclusion relation is a totally ordered set.
Now, if we have a space which is not $N_1$ we may not able to use this inductive construction, so my question is if there is some other technique we can use for a (possible) non-countable neighborhood basis of a point or if it is proven that is not possible to construct such a basis.
Thanks