Exercise 14, § 2

a) Let $X$ be an accessible topological space (Chapter I, § 8, Exercise 1). Show that the following properties are equivalent:

$\alpha$ Every sequence of points of $X$ has a cluster point. $\beta$ No infinite discrete subspace of $X$ is closed. $\gamma$ Every countable open covering of $X$ contains a finite open covering $X$. $\delta$ Given any infinite open covering $\mathfrak{R}$ of $X$, there exists an open covering $\mathscr{G} \subset \mathfrak{R}$ of $X$, distinct from $\mathfrak{R}$.

A topological space $X$ is said to be countably compact if it is Hausdorff and has the above properties.

b) A sequence of points in a countably compact space is convergent if and only if it has a unique cluster point.

c) Every closed subspace of a countably compact space is countably compact. Conversely, if $X$ is Hausdorff and if every point of $X$ has a countable fundamental system of neighbourhoods, then every countably compact subspace of $X$ is closed in $X$.

d) Let $f$ be a continuous mapping of a countably compact space $X$ into a Hausdorff space $X'$. Then $f(X)$ is a countably compact subspace of $X'$.

e) Let $X$ be a countably compact space, every point of which has a countable fundamental system of neighbourhoods. Then every sequence of points of $X$ has a convergent subsequence.

f) Let $(X_n)$ be a countable sequence of topological spaces, in each of which every point has a countable fundamental system of neighbourhoods.

The product space $\prod_{n=1}^\infty X_n$ is then countably compact if and only if each of the spaces $X_n$ is countably compact. [Use e) and Chapter I, § 6, Exercise 16.]

g) A countably compact space in which every point has a countable fundamental system of neighbourhoods is regular.

h) If a countably compact space has a countable base of open sets, it is compact and therefore metrizable.

i) Every lower semi-continuous real-valued function on a countably compact space attains its greatest lower bound. In particular, every countably compact space is pseudo-compact ($\S \ 1,$ Exercise 21) [cf. $\S \ 4,$ Exercise 26 b)].

j) Show that the property $\delta)$ of a) implies the following:

$\zeta)$ Every point-finite ($\S \ 4,$ no. 3) open covering of $X$ contains a finite open covering of $X$. (Argue by contradiction.) Conversely, a regular space which satisfies $\zeta)$ is countably compact [show that $\zeta)$ then implies $\beta)$].