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?
