1st December 2004
Munkres
§
34
Ex. 34.1.
We are looking for a nonregular 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 endpoints, 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 halfopen 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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 Fall '08
 Brown
 Topology, Sets, UK, Topological space, compact Hausdorff space, ∞

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