# T4 and first countable topology that is non metrizable

Does anyone know any example of such topology?

• Does T4 imply Hausdorff for you? – Miha Habič Aug 22 '13 at 15:24
• @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. – Cameron Buie Aug 22 '13 at 16:00
• Yes, T4=> Hausdorff and regularity – Fagner Santana Aug 22 '13 at 16:39
• @Cameron: It should, but unfortunately the opposite convention is also found, so it’s usually a good idea to ask. – Brian M. Scott Aug 22 '13 at 16:54
• @Brian: That is unfortunate. I'll have to bear that in mind. – Cameron Buie Aug 23 '13 at 14:01

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.