A non metric first countable topological space

Question

Every metric space is first countable, but what about the converse? Does it always hold? If not, can anyone give a counterexample? Thanks

score 5 · Answer 1 · answered Apr 10 '16 at 18:24

5

The converse is false. Take the Sierpinski space $X = \{0,1\}$ with topology given by $\tau = \{ \varnothing, \{0\}, \{0,1\} \}$. This topology is first-countable but it is not Hausdorff, hence not metrizable.

answered Apr 10 '16 at 18:24

Ivo Terek

77,665

hmakholm left over Monica · Accepted Answer · 2016-04-10T18:37:20.753

4

The long line is locally homeomorphic to $\mathbb R$ (and so first countable) but is not metrizable.

Less exotically, the lower limit topology on $\mathbb R$ is also first countable but not metrizable.

edited Apr 10 '16 at 18:37

answered Apr 10 '16 at 18:21

hmakholm left over Monica

286,031

A non metric first countable topological space

2 Answers2