Normality is weakly hereditary

This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
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This article gives the statement, and possibly proof, of a basic fact in topology.

Statement

Verbal statement

Any closed subset of a normal space is also normal, in the subspace topology.

Definitions used

Normal space

Further information: normal space

Subspace topology

Further information: subspace topology

Proof

Proof outline

Note that the property of being a T1 space is certainly hereditary to all subspaces, so we only need to check the separation of closed subsets.

We proceed as follows:

• Pick two closed subsets inside the subspace
• Observe, using the fact that a closed subspace of a closed subspace is closed, that both of them are closed in the whole space
• Separate them by disjoint open sets in the whole space
• Intersect these with the subspace, and use the definition of subspace topology to conclude that we have a separation by disjoint open sets in the subspace

References

Textbook references

• Topology (2nd edition) by James R. Munkres, ^{More info}, Page 205, Exercise 1, Chapter 4, Section 32