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How to prove that a map from $X$ to a subset $A$ of $Y$, with the subspace topology is continuous (inverse image of open sets are open) iff the composite $X \to A \subseteq Y$ is continuous?

What is meant by the composite $X \to A \subseteq Y$?

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The composite in question is the composite with the inclusion map $i: A \hookrightarrow B$ (which takes a point in $A$ to itself). To show the result, think about what $i^{-1}(C)$ is, for $C \subseteq B$, and how this might be related to the subspace topology on $A$.