Topology from subspace metric equals subspace topology

Statement

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 Subspace metric (?) on $Y$, by restriction. Thus, there are two possible topologies we can put on $Y$:

• The 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.

Proof

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