Exercise 2, § 5

a) Let X be a regular space and A a subset of X. Suppose that every sequence of points of A has a limit point in X and that there exists a metrizable topology T on X which is coarser than the given topology T_0. Show that the closure Ā of A in X is a compact metrizable space (arguing by reductio ad absurdum, show that the topologies induced by T and T_0 on Ā are identical). b) Let E be a Hausdorff locally convex space, which is the union of a sequence (E_n) of metrizable vector subspaces for the topology induced by that of E. Show that, for a subset A of E to be relatively compact in E, it is necessary and sufficient that every sequence of points of A has a limit point in E. (If (W_{mn}) (for m ≥ 1) is a fundamental system of convex neighbourhoods of 0 in E_n which are open in E_n, let U_{mn} be a convex neighbourhood of 0 in E such that E_n ∩ U_{mn} = W_{mn} (II, p. 33, lemma 2); on E consider the topology for which the U_{mn} (m ≥ 1, n ≥ 1) form a fundamental system of neighbourhoods of 0). c) Extend Šmulian’s theorem to a strict inductive limit space (II, p. 33) of Fréchet spaces.