# 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. MunkresMore info, Page 190, Chapter 4, Section 30 (formal definition)