Exercise 21, § 1

a) Let $X$ be a topological space. Show that the following two properties are equivalent: $\alpha)$ every finite continuous real-valued function on $X$ is bounded; $\beta$) every bounded continuous real-valued function on $X$ attains its bounds. $X$ is said to be pseudo-compact if it has these properties. If $X$ is pseudo-compact and if $f$ is any continuous mapping of $X$ into a topological space $X'$, then $f(X)$ is pseudo-compact.

b) Consider the following two properties of a topological space $X$: $\gamma)$ if $(U_n)$ is any countable open covering of $X$, then $X$ is the union of a finite number of the closures $\overline{U}_n$; $\delta)$ every countable filter base on $X$ which is formed of open sets has at least one cluster point. Show that $\gamma)$ and $\delta)$ are equivalent and that they imply that $X$ is pseudo-compact. In particular, every absolutely closed topological space (Chapter I, § 9, Exercise 19) is pseudo-compact. If $X$ has properties $\gamma)$ and $\delta)$, then so does every subspace of $X$ which is the closure of an open set of $X$ [cf. § 4, Exercise 26 b)].

c) Consider the following property of a topological space $X$: $\zeta)$ every locally finite open covering of $X$ is finite. Show that $\zeta) \Longrightarrow \gamma)$. [If $(U_n)$ is an increasing sequence of open subsets of $X$ which cover $X$ and are such that $U_{n+1} \not\subset \overline{U}_n$, consider the covering formed by the sets $U_{n+1} \cap \complement \overline{U}_n$ and the complement of a sequence $(a_n)$ such that

If $X$ is regular, show that $\gamma) \Longrightarrow \zeta)$. [Let $(U_\alpha)$ be an infinite, locally finite, open covering of $X$. Define by induction a sequence $(\alpha_n)$ of indices, a sequence $(x_n)$ of points of $X$, and for each $x_n$ two open neighbourhoods $V_n$ and $W_n$ of $x_n$ such that: (i) we have $\overline{V}_n \subset W_n$, $W_n \subset U_{\alpha_n}$, and $W_n$ meets only a finite number of sets $U_\alpha$; (ii) $U_{\alpha_n}$ meets none of the $W_k$ with indices $k < n$. Now consider the covering formed by the $W_n$ and the complement of the union of the $\overline{V}_n$.]

d) In a completely regular space $X$ the properties $\alpha), \beta), \gamma), \delta)$ and $\zeta)$ are all equivalent, and are equivalent to the following property: $\theta)$ $X$ is precompact in any uniformity compatible with its topology. [To show that $\alpha) \Longrightarrow \zeta)$, note that if $(U_n)$ is a countably infinite, locally finite, open covering of $X$ and if $a_n \in U_n$, then there is a continuous real-valued function $f$ on $X$ such that $f(a_n) = n$ for each $n$ [use (O_{IV})]. To show that $\theta) \Longrightarrow \alpha)$, use Exercise 5 and observe that in a precompact space every uniformly continuous real-valued function is bounded. To show that $\gamma) \Longrightarrow \theta)$, note that if $X$ is not precompact with respect to a uniformity $\mathscr{U}$, then there is a symmetric entourage $V$ of $\mathscr{U}$ and a sequence $(x_n)$ of points of $X$ such that no two of the sets $V(x_n)$ intersect.]

e) If $X$ is a completely regular pseudo-compact space which is complete with respect to some uniformity compatible with its topology, then $X$ is compact.

f) On a completely regular pseudo-compact space $X$, the universal uniformity (Exercise 5) is induced by the uniformity of the Stone-Čech compactification of $X$, and is the unique uniformity, compatible with the topology of $X$, for which all the continuous mappings of $X$ into $[0, 1]$ are uniformly continuous.