First-countable space

Symbol-free definition
A topological space is said to be first-countable if for any point, there is a countable basis at that point.

Definition with symbols
A topological space $$X$$ is said to be first-countable if for any $$x \in X$$, there exists a countable collection $$U_n$$ of open sets around $$x$$ such that any open $$V \ni x$$ contains some $$U_n$$.

Stronger properties

 * Second-countable space
 * Metrizable space
 * Locally metrizable space

Weaker properties

 * Compactly generated space:
 * Sequential space

Metaproperties
Any subspace of a first-countable space is first-countable. We can take, for our new basis at any point, the intersection of the old basis elements with the subspace.

Any countable product of first-countable spaces is first-countable.

If every point has a neighbourhood which is first-countable, then the whole topological space is first-countable.

Textbook references

 * , Page 190 (formal definition)
 * , Page 39 (formal definition)