1st December 2004Munkres§34Ex. 34.1.We are looking for a non-regular Hausdorff space. By Example 1 p. 197,RK[p. 82]is such a space.Indeed,RKis Hausdorff for the topology is finer than the standard topology[Lemma 13.4].RKis 2nd countable for the sets (a, b) and (a, b)-K, where the intervals haverational end-points, constitute a countable basis.RKis 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 Urysohnmetrization 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 topologyR[p. 82] is completelynormal [Ex 32.4], 1st countable, Lindel¨of, has a countable dense subset [Example 3, p. 192], butis not metrizable [Ex 30.6].Ex. 34.3.We characterize the metrizable spaces among the compact Hausdorff spaces.Theorem 1.LetXbe a compact Hausdorff space. ThenXis metrizable⇔Xis2nd countableProof.⇒: Every compact metrizable space is 2nd countable [Ex 30.4].
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