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)
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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