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Let $(X,\tau)$ be a topological space and $A$ a nonempty set. Let $X^A$ be the set of all mappings from $A$ to $X$. It is clear to me that the topology of pointwise convergence on $X^A$ is just the usual product topology on $X^A$ viewed as a Cartesian product.

However, let $\mathcal M$ be a nonempty proper subset of $X^A$. For example, attention is restricted on such mappings from $A$ to $X$ that satisfy some additional properties (say, continuity, when $A$ is topologized in some way), so that we no longer consider the set of all mappings from $A$ to $X$, just the set of some mappings from $A$ to $X$.

My question is: What is the proper interpretation of the notion of the “topology of pointwise convergence on $\mathcal M$?” Is it just the relative topology on $\mathcal M$ induced by the usual topology of pointwise convergence on $X^A$? Or is it usually defined some other way that differs from the relative topology?

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Yes, the topology of pointwise convergence on a subset $\mathcal{M} \subset X^A$ is the subspace topology induced on $\mathcal{M}$ by the product topology on $X^A$.

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I thought so, it does make sense, just wanted to make sure. Thank you very much for the quick answer! –  triple_sec Dec 13 '13 at 20:40

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