Normality is weakly hereditary

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

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

Textbook references

 * , Page 205, Exercise 1, Chapter 4, Section 32