Exercise 9, § 4

Let X be a locally compact space which carries a uniformity compatible with its topology and is such that there is an entourage V with the property that V(x) is relatively compact in X for all x ∈ X. a) Show that X is complete with respect to this uniformity. b) Let U be a symmetric entourage of X such that $\hat{U} \subset V$. Show that if A is relatively compact in X, then so is U(A). c) Show that X is paracompact [use b), Exercise 7 and Chapter I, § 9, no. 10, Theorem 5]. d) Let W be a closed symmetric entourage of X such that $\hat{W} \subset V$. Show that if A is closed in X then so is W(A) (use Exercise 8).