Metrizability is hereditary

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

Facts used

  1. Topology from subspace metric equals subspace topology

Proof

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