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MATH 730: PROBLEM SET 2 WILLIAM GOLDMAN (1) (a) Let X be a locally compact Hausdorff space. Then the intersection of an open subset of X and a closed subset of X is locally compact. (b) Let X be Hausdorff and Y ⊂ X be locally compact. Show that Y is the intersection of an open subset of X and a closed subset of X. (2) Prove or disprove: The continuous image of a locally compact space is locally compact. (3) (Bredon §I.11.1, p.31) Recall that the quasicomponents of a topological space X are the equivalence classes under the equivalence relation f x ∼ y :⇐⇒ f (x) = f (y)∀X → − D where D is a discrete space. Prove or disprove: (a) If X is compact and Hausdorff, then the quasi-components are the connected components of X.a (b) Same if X is only assumed to be compact. (c) Same if X is only assumed to be Hausdorff. (4) Find an example of a continuous open map which is not cloaed, and a continuous closed map which is not open. Must every continuous map be either open or closed? (5) Prove or disprove: If a topological space X satisfies X ≈ X ×X, then X = ∅. Date: 14 September 2006. 1