1. Let (X, d) be a locally compact separable metric space. Show that its one-point compactification is metrizable.
2. Let X be any noncompact metric space, considered as a subset of its Tychonoff-C? ech compactification K . Let y ∈ K \X . Show that K is not metrizable by showing that there is no sequence xn ∈ X with xn → y in K . Hint: If xn → y, by taking a subsequence, assume that the points xn are all different. Then, {x2n }n≥1 and {x2n-1}n≥1 form two disjoint closed sets in X . Apply Urysohn's Lemma (2.6.3) to get a continuous function f on X with f (x2n ) = 1 and f (x2n-1) = 0 for all n. So {xn } cannot converge in K to y.