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 nsphere 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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 Spring '08
 Ogle,C
 Topology

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