Metrizable space

Symbol-free definition
A topological space is said to be metrizable if it occurs as the underlying topological space of a metric space via the naturally induced topology with the basis open sets being the open balls with center in the space and finite positive radius.

Metaproperties
Any subspace of a metrizable space is metrizable. In fact, the subspace topology coincides with the topology induced from the metric obtained on the subset ,by restricting the metric from the whole space.

A finite product of metrizable spaces is again metrizable. In fact, we can take the metric as, say, the sum of metric distances in each coordinate. More generally, we could use any of the $$L^p$$-norms ($$1 \le p \le \infty$$) to combine the individual metrics.

Textbook references

 * , Page 120, Chapter 2, Section 20 (formal definition, along with metric space)