First-countable space

Definition

Let $X$ be a topological space. The space $X$ is first-countable when each point $x \in X$ has a sequence $(N_n)_{n \ge 1}$ of neighbourhoods with this property: each neighbourhood of $x$ contains some $N_n$.

Remarks

Such a sequence is a countable neighbourhood basis at $x$. One can choose the sets $N_n$ open, and also decreasing. Each metric space is first-countable: take $N_n = B(x, 1/n)$, as in open balls form a basis. Each second-countable space is first-countable. In a first-countable space, a point is in the closure of a set $A$ exactly when it is a limit of a sequence in $A$.

Explanations

L1 human

At each point, you can shrink to the point along a countable list of neighbourhoods. In a metric space, the balls of radius $1/n$ do this.