325 HW 07 - MAT 325: Topology Professor Zoltan Szabo...

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MAT 325: Topology Professor Zoltan Szabo Problem Set 7 Rik Sengupta rsengupt@princeton.edu April 7, 2010 1. Munkres, p. 218, problem 1 Give an example showing that a Hausdorff space with a countable basis need not be metrizable. Solution. R K suffices as an example. By Example 1 on p. 197, R K [as defined on p. 82] is a space that is non-regular and Hausdorff. Indeed, R K is Hausdorff, for the topology is finer than the standard topology [see Lemma 13.4]. Furthermore, R K is 2nd countable, for the sets ( a,b ) and ( a,b ) - K , where the intervals have rational end-points, constitute a countable basis. However, R K is not metrizable, for it is not even regular [by Example 1, p. 197]. Here, recall that a metrizable space is necessarily T 1 and regular, which gives rise to the contradiction. ± 2. Munkres, p. 218, problem 3 Let X be a compact Hausdorff space. Show that X is metrizable if and only if X has a countable basis. Solution. We present two solutions: Solution 1. (= :) Assume X is metrizable. Now, X is Lindel¨of because it is compact, and metrizable Lindel¨of spaces are second countable. ( =:) Assume X is second countable. Since X is compact and Hausdorff, it is normal, and thus regular as well. Urysohn’s metrization theorem then shows that X is metrizable. ± Solution 2. We first prove a simple lemma. Lemma. Every compact metrizable space is second-countable. Proof. Let X be a compact metrizable space, and let d be a metric on X that induces the topology on X . For each n Z + , let A n be an open covering of X with 1 /n -balls. By the compactness of X , there exists a finite subcovering A n . So, B = S n Z + A n is countable, being a countable union of finite sets. Notice that B is a basis. Let U be an open set in X and x U . By definition of the metric topology, there exists some ± > 0 such that B d ( x,± ) U . Choose N Z + such that 2 /N < ± . Since A n covers X , this means there exists some B d ( y, 1 /N ) containing x . If z B d ( y, 1 /N ), then 1
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d ( x,z ) d ( x,y ) + d ( y,z ) 1 /N + 1 /N = 2 /N < ±, i.e. z B d ( x,± ), hence B d ( y, 1 /N ) B d ( x,± ) U . It follows that B is a basis. ± Proof of Problem. We characterize the metrizable spaces among the compact Hausdorff spaces. Theorem. Let X be a compact Hausdorff space. Then X is metrizable iff X is 2nd countable. Proof. (= :) Every compact metrizable space is 2nd countable, by the lemma above. ( =:) Every compact Hausdorff space is normal, from Theorem 32.3. Every 2nd countable normal space is metrizable by the Urysohn metrization theorem, by Theorem 34.1. Now, we may also characterize the metrizable spaces among 2nd countable spaces.
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This note was uploaded on 05/07/2010 for the course MAT 325 taught by Professor Zoltánszabó during the Fall '09 term at Princeton.

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325 HW 07 - MAT 325: Topology Professor Zoltan Szabo...

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