Second-countable space

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

Definition

A topological space is termed second-countable if it satisfies the following equivalent conditions:

• It admits a countable basis, i.e., a countable collection of open subsets that form a basis for the topology.
• It admits a countable subbasis, i.e., a countable collection of open subsets that form a subbasis for the topology.

Relation with other properties

Stronger properties

• Separable metrizable space
• Polish space
• Sub-Euclidean space

Weaker properties

• Hereditarily separable space
• Separable space: For proof of the implication, refer Second-countable implies separable and for proof of its strictness (i.e. the reverse implication being false) refer Separable not implies second-countable
• First-countable space: For proof of the implication, refer Second-countable implies first-countable and for proof of its strictness (i.e. the reverse implication being false) refer First-countable not implies second-countable
• Lindelof space: For proof of the implication, refer Second-countable implies Lindelof and for proof of its strictness (i.e. the reverse implication being false) refer Lindelof not implies second-countable

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 second-countable space is second-countable. For full proof, refer: Second-countability is hereditary

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 190, Chapter 4, Section 30 (formal definition)