A Topological space $X$ is first countable in a point $x \in X$ if there exists a countable system of Neighbourhoods $U_1, U_2, U_3, ...$ such that given an arbitrary nhbd $U$ of $x$, there exists $n \in \mathbb{N}$ such that $U_n \subset U$.

We can always assume that $U_1 \supset U_2 \supset U_3 \supset ...$ (proof by taking intersections of previous $U_i$). In this form, the system is called a foundamental system of neighbourhoods of $x$

Applications to the Sequence lemma

$X$ is first countable if it is first countable in every point. Metrizable implies first countable