Second-countability is hereditary

From Topospaces
Jump to: navigation, search
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

Statement

Property-theoretic statement

The property of topological spaces of being a second-countable space satisfies the metaproperty of topological spaces of being hereditary.

Verbal statement

Any subspace of a second-countable space is second-countable under the subspace topology.

Definitions used

Second-countable space

Further information: Second-countable space

A topological space is termed second-countable if it admits a countable basis.

Subspace topology

Further information: Subspace topology

Given a topological space X and a subspace A, with a basis \{ B_i \}_{i \in I} for X, the subspace topology on A is defined as a topology with basis B_i \cap A.

Proof

Given: A second-countable space X with countable basis B_n, n \in \mathbb{N}. A subspace A of X

To prove: A has a countable basis.

Proof: By the definition of subspace topology, the sets B_n \cap A form a basis for the subspace topology on A. This is a countable basis for A.

References

  • Topology (2nd edition) by James R. Munkres, More info, Page 191, Chapter 4, Section 30