Exercise 21, § 4

Now prove that for a completely regular space $X$, the following four conditions are equivalent:

a) $X$ is replete (X, p. 59, Exercise 17 b)). b) $X$ is the intersection of the subsets of $\beta X$ which contain it and which are countable unions of compact sets.

c) $\beta X - X$ is a union of closed sets each of which is a countable intersection of open sets.

d) $X$ is homeomorphic to a dense subset of a compact space $K$, which is the intersection of a family of subsets of $K$ which are countable unions of compact sets.