Topology from subspace metric equals subspace topology

Statement with symbols
Suppose $$(X,d)$$ is a metric space. Then, we can consider the induced topology on $$X$$ from the metric.

Now, consider a subset $$Y$$ of $$X$$. The metric on $$X$$ induces a fact about::subspace metric on $$Y$$, by restriction. Thus, there are two possible topologies we can put on $$Y$$:


 * The fact about::subspace topology from the topology induced by the metric on $$X$$
 * The induced topology from the subspace metric on $$Y$$

These two topologies are the same.