Let $f: X \to Y$ be a continuous map between topological spaces, and let $A \subset X$ carry the subspace topology.
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1. The **restriction** $f|_A: A \to Y$, defined by $f|_A(a) = f(a)$, is continuous.
2. If $f(A) \subset B$ for some subspace $B \subset Y$, then the **corestriction** $f|_A^B: A \to B$ (same rule as $f$, but with codomain $B$) is continuous.