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 finite or countable basis, i.e., a finite or countable collection of open subsets that form a basis for the topology.
• It admits a finite countable subbasis, i.e., a finite or countable collection of open subsets that form a subbasis for the topology.
• The weight of the space is either finite or countable.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
separable metrizable space
Polish space
Sub-Euclidean space
second-countable T1 space

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
hereditarily separable space
separable space		second-countable implies separable	separable not implies second-countable
first-countable space		second-countable implies first-countable	first-countable not implies second-countable
Lindelof space		second-countable implies Lindelof	Lindelof not implies second-countable
compactly generated space		via first-countable	via first-countable	|FULL LIST, MORE INFO

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)