First-countable space

This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

Definition

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 $U_{n}$.

Relation with other properties

Stronger properties

• Second-countable space
• Metrizable space

Weaker properties

• Sequential space

Metaproperties

Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces

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. For full proof, refer: First-countability is hereditary

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