Remark 2

There are examples of normal spaces which are neither metrizable nor locally compact (see § 5, Exercise 16).

By $(O'_V)$, every closed subset of a normal space is a normal subspace; but this is not always the case for an arbitrary subset of a normal space.

For example, a completely regular space which is not normal is homeomorphic to a subspace of a compact space (§ 1, no. 5, Proposition 3), and the latter is normal.

Finally we record that the product of two normal spaces is not necessarily normal (see Exercise 9 and § 5, Exercise 16).