Second-countability is hereditary
This article gives the statement, and possibly proof, of a topological space property (i.e., second-countable space) satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces)
View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions
Get more facts about second-countable space |Get facts that use property satisfaction of second-countable space | Get facts that use property satisfaction of second-countable space|Get more facts about subspace-hereditary property of topological spaces
Further information: Second-countable space
Further information: Subspace topology
Given a topological space and a subspace , with a basis for , the subspace topology on is defined as a topology with basis .
Given: A second-countable space with countable basis . A subspace of
To prove: has a countable basis.
Proof: By the definition of subspace topology, the sets form a basis for the subspace topology on . This is a countable basis for .
- Topology (2nd edition) by James R. Munkres, More info, Page 191, Chapter 4, Section 30