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
• Locally metrizable space

Weaker properties

• Compactly generated space: For full proof, refer: First-countable implies compactly generated
• 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.

Local nature

This property of topological spaces is local, in the sense that the topological space satisfies the property if and only if every point has an open neighbourhood which satisfies the property

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

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 190 (formal definition)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 39 (formal definition)