Second-countable space

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.

Metaproperties
Any subspace of a second-countable space is second-countable.

Textbook references

 * , Page 190, Chapter 4, Section 30 (formal definition)