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