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hmw6.1 - Math 655 — Homework E 1 Let f X — Y be a...

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Unformatted text preview: Math. 655 — Homework # E 1} Let f : X —} Y be a closed map {ie. if A C X is closed. then so is HAD with the additional property that the inverse image of any point is compact. Show that f_l[H} is compact whenever If is compact. Can you remove the condition the f is closed? 2) A space X is said to be locally compact if for any 3: E X we can find an open set U containing :1: such that E is compact. Suppose that X is locally compact and Hausdorff but not compact. Form a new space by adding one extra point denoted cc: to X and specifying that the open sets of X LJ one are the open sets of X and sets of the form (X — H) U DO, where H is compact in X. Show that the result is a topological space which is also Hausdorff. and that X is dense in X Ll DD. (X LJ cc: is called the one—point compocﬁﬁcction of X.) What happens if X is already compact? 3} Show that R“ U can is homeomorphic to the unit n-sphere S”:{[Il,xg,...,xn+1)|:c§+33+...+:c:+1= 1} 4} Let X and Y be locally compact Hausdorff spaces and f : X —I- Y an onto map. Show that f extends to a map ...
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