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 admits a countable basis.

Relation with other properties

Stronger properties

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

Weaker properties

• Hereditarily separable space
• Separable space
• First-countable 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 second-countable space is second-countable.

References

Textbook references

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