Exercise 7, § 4

A subset $A$ of a uniform space $X$ is said to be bounded if for each entourage $V$ of $X$ there is a finite set $F$ and an integer $n > 0$ such that $A \subset^n V(F)$.

a) The union of two bounded subsets is bounded. The closure of a bounded subset is bounded. Every precompact subset is bounded.

b) If $f : X \to Y$ is uniformly continuous, then the image under $f$ of every bounded subset of $X$ is a bounded subset of $Y$.

c) In a product of non-empty uniform spaces, a set is bounded if and only if each of its projections is bounded.

d) For each symmetric entourage $V$ of $X$, let $R_V$ denote the union of the sets $V^n$ as $n$ runs through the integers $> 0$. Show that $R_V$ is both open and closed in $X \times X$ and that, in the notation of no. 4,

for each $x \in X$. Suppose that $R_V = X \times X$ for each symmetric entourage $V$ of $X$ (this will be the case in particular if $X$ is connected); show that a set $A \subset X$ is bounded if and only if, for each symmetric entourage $V$ of $X$ and each $x_0 \in X$, there is an integer $n > 0$ such that $A \subset^n V(x_0)$.