If \(S\) is a subspace of a topological space \(X\), the inclusion map \(\iota_S\colon S\hookrightarrow X\) is continuous function .
Proof
Suppose \(\id:S\to S\) is the identity map. According to (0x678ea995)
it is continuous. Applying the characteristic property of the subspace topology shows that \(\iota_S=\iota_S\circ \id\) is also continuous.