# Can a topology induce a metric?

I'm learning the basics of topology and I feel the need to ask a vague question. I'm familiar with metrics inducing topologies, but is it possible for the inverse to be true?

I know multiple metrics can induce a single topology so my intuition would say that a topology would induce an infinite number of metrics. I probably have a misunderstanding of how these things work.

Thanks!

• In general, the answer is no. Every metric space is Hausdorff but there are topological spaces which are not. See en.wikipedia.org/wiki/Metrization_theorem – mfl Oct 21 '18 at 22:04
• Also, each metric induces exactly one topology: the topology whose open sets are exactly those defined by the metric. – Ricky Tensor Oct 21 '18 at 22:06
• Thanks! I wrote down my question slightly wrong, so I fixed it with an edit. – jermanmao Oct 21 '18 at 22:40

## 1 Answer

First of all, every metric induces one topology: the topology for which the open sets are the unions of open balls.

On the other hand, many topologies are induced by no matric. Take, for instance, the lower limit topology on $$\mathbb R$$, which is the topology generated by the intervals of th type $$[a,b)$$.