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Transcript

Compactness (1) Let f : X → Y be continuous and X compact. Prove that f (X) is compact. (i.e. the continuous image of a compact space is compact.) (2) Let X be compact and A ⊂ X closed. Prove that A is compact. (3) Classify all compact discrete topological spaces, up to homeomorphism. (4) Prove the Bolzano-Weierstrass property: An infinite subset of a compact space must have a limit point. (Hint: Prove instead the contrapositive: any subset of a compact space which has no limit point is finite.) 1