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(a) Suppose K is a compact subset of a metric space (X, d) and x є X but x K Show that there exist disjoint, open subsets of Gi and G2 of (X, d) such that r E Gi and KG2. (Hint: Use the version of co...
Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R. Suppose that, for each the sequence (fe(x))ke N 1s a monotonic sequence which converges to (x). Show that r є X, k)kEN Converges to j uniformly. Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R....