# Metrizability is hereditary

From Topospaces

This article gives the statement, and possibly proof, of a topological space property (i.e., metrizable space) satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces)

View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions

Get more facts about metrizable space |Get facts that use property satisfaction of metrizable space | Get facts that use property satisfaction of metrizable space|Get more facts about subspace-hereditary property of topological spaces

## Statement

If is a metrizable space, and is a subset of given the subspace topology, then is also a metrizable space.

## Facts used

## Proof

Suppose is a metric on inducing the given topology on . By fact (1), the subspace metric on induces the subspace topology on . Thus, the subspace topology is induced by a metric, hence is metrizable.