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

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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. Theorem 1. Let X be a compact Hausdorff space. Then X is metrizable X is 2 nd countable Proof. : Every compact metrizable space is 2nd countable [Ex 30.4].
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