S34 - 1st December 2004 Munkres 34 Ex. 34.1. We are looking...

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Unformatted text preview: 1st December 2004 Munkres 34 Ex. 34.1. We are looking for a non-regular Hausdorff space. By Example 1 p. 197, R K [p. 82] is such a space. Indeed, R K is Hausdorff for the topology is finer than the standard topology [Lemma 13.4]. 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. R K is not metrizable for it is not even regular [Example 1, p. 197]. Conclusion : The regularity axiom can not be replaced by the Hausdorff axiom in the Urysohn metrization theorem [Thm 34.1]. Ex. 34.2. We are looking for 1st but not 2nd countable space. By Example 3 p. 192, R ` [p. 82] is such a space. Indeed, the Sorgenfrey right half-open interval topology R ` [p. 82] is completely normal [Ex 32.4], 1st countable, Lindel of, has a countable dense subset [Example 3, p. 192], but is not metrizable [Ex 30.6]. Ex. 34.3. We characterize the metrizable spaces among the compact Hausdorff spaces....
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This note was uploaded on 01/12/2011 for the course MATH 110 taught by Professor Brown during the Fall '08 term at Arizona Western College.

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