4
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Does anyone know any example of such topology?

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    $\begingroup$ Does T4 imply Hausdorff for you? $\endgroup$ – Miha Habič Aug 22 '13 at 15:24
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    $\begingroup$ @Miha: $T_n$ for $n>2$ should always imply Hausdorff. "Normal" need not imply Hausdorff, though, unless the definition of "normal" assumes that finite sets are closed. $\endgroup$ – Cameron Buie Aug 22 '13 at 16:00
  • $\begingroup$ Yes, T4=> Hausdorff and regularity $\endgroup$ – Fagner Santana Aug 22 '13 at 16:39
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    $\begingroup$ @Cameron: It should, but unfortunately the opposite convention is also found, so it’s usually a good idea to ask. $\endgroup$ – Brian M. Scott Aug 22 '13 at 16:54
  • $\begingroup$ @Brian: That is unfortunate. I'll have to bear that in mind. $\endgroup$ – Cameron Buie Aug 23 '13 at 14:01
10
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The Sorgenfrey Line is $T_4$ and first-countable. It cannot be metrizable since it has a countable dense subset $\Bbb Q$ but has no countable base, that means it is separable without being second-countable.

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    $\begingroup$ Alternatively, it cannot be metrizable because its square isn’t normal. $\endgroup$ – Brian M. Scott Aug 22 '13 at 16:27
  • $\begingroup$ Ok, the sorgenfrey line, always a good counter-example. It is perfectly normal too. $\endgroup$ – Fagner Santana Aug 22 '13 at 16:54
7
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The first uncountable ordinal with the order topology is normal, first countable and even locally metrizable - but it is not metrizable, since it is non-compact despite the fact that it is sequentially compact

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