This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
Definition with symbols
A topological space is said to be first-countable if for any , there exists a countable collection of open sets around such that any open contains some .
Relation with other properties
- Compactly generated space: For full proof, refer: First-countable implies compactly generated
- Sequential space
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.
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.
- Topology (2nd edition) by James R. MunkresMore info, Page 190 (formal definition)
- Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. ThorpeMore info, Page 39 (formal definition)