# 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

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 $$[a,b)$$.