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Exercise 2.3.1 (Prove Theorem 2.3.1.)
Prove Theorem 2.3.1. Theorem 2.3.1 (Continuous maps preserve compactness). Let f : X → Y be a continuous map from one metric space (X,dX) to another (Y, dY ). Let K X be any compact subset of X. Then the image f(K) := f(x) : x K of K is also compact.