 Metrizable space

A topological space is a metrizable space if some metric on its set generates exactly its given topology. The choice of metric need not be unique. For example, $d(x,y)=|x-y|$ and $\min (1,|x-y|)$ generate the same topology on the real numbers even though one is unbounded and the other bounded.

Metrizability lets one use metric balls and sequential arguments to study the space. Every metric topology is Hausdorff and first countable, so a topology lacking either property is not metrizable. Conversely those two properties alone are not a complete metrization criterion; additional hypotheses occur in the standard metrization theorems.

A **metrizable space** is a topological space that can be endowed with a metric (or distance function) such that the topology induced by this metric is the same as the original topology of the space.