MATH527_HW4 - Serge Ballif MATH 527 Homework 4 Problem 1...

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Serge Ballif MATH 527 Homework 4 September 28, 2007 Problem 1. Show that every locally compact Hausdorff space is completely regular. Let X be locally compact Hausdorff. Then X satisfies the conditions necessary for a one-point compactification. That is, there exists a compact Hausdorff set Y containing X as a subspace. In particular, Y is normal. Therefore, Y is com- pletely regular. The space X is completely regular as a subspace of a completely regular space. Problem 2. A space X is locally metrizable if each point x X has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space X is metrizable if and only if it is locally metrizable. ( ) Any metrizable space is locally metrizable, since a subspace of a metric space is a metric space. ( ) Let X be compact, Hausdorff, and locally metrizable. X is normal (and hence, regular) as a compact Hausdorff space. If X has a countable basis, then we can invoke the Urysohn metrization theorem to show that X is metrizable. To show that X has a countable basis, we note that local metrizability of X means that for each x in X , there is an open neighborhood U x of x which is metrizable. Then the set U c x is closed in X and is disjoint from x . Therefore, by regularity of X there are disjoint open sets V x ⊃ { x } and W x U c x . The set V x is a closed compact subset that is contained in U x . We can repeat this procedure for each x in X . Then since X is compact, there exists a finite subcover of sets V x 1 , V x 2 , . . . , V x k . Each space V x i is compact as a closed subspace of a compact space, and is metrizable as a subset of U x i . Hence, V x i is 2nd countable with basis B i . The collection
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