# S32 - 1st December 2004 Munkres 32 Ex 32.1 Let Y be a...

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1st December 2004 Munkres § 32 Ex. 32.1. Let Y be a closed subspace of the normal space X . Then Y is Hausdorﬀ [Thm 17.11]. Let A and B be disjoint closed subspaces of Y . Since A and B are closed also in X , they can be separated in X by disjoint open sets U and V . Then Y U and V Y are open sets in Y separating A and B . Ex. 32.3. Look at [Thm 29.2] and [Lemma 31.1]. By [Ex 33.7], locally compact Hausdorﬀ spaces are even completely regular. Ex. 32.4. Let A and B be disjoint closed subsets of a regular Lindel¨ of space. We proceed as in the proof of [Thm 32.1]. Each point a A has an open neighborhood U a with closure U a disjoint from B . Applying the Lindel¨ of property to the open covering { U a } a A ∪ { X - A } we get a countable open covering { U i } i Z + of A such that the closure of each U i is disjoint from B . Similarly, there is a countable open covering { V i } i Z + of B such that the closure of each V i is disjoint from A . Now the open set S
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