Test continuity with projections If $Y$ and $X_i$ are topological spaces and $\prod_{i\in I}X_i$ has the product topology, how can I test continuity of the map $f:Y\rightarrow \prod_{i\in I}X_i$ using the projections?
This question is related to a problem I posted here: $f: \mathbb{R}\rightarrow \mathbb{R}^{\mathbb{R}}: x\mapsto (e^{t\sin(x)})_{ t\in \mathbb{R}}$
k.stm has solved this using the product topology. My problem is that I cannot see the connection between projections and this proof.
I'm posting this here as a general question because I want to investigate the general approach to such problems.
 A: Given a family $X$ of sets indexed by $I$ and a family $\mathscr{T} \in \displaystyle\prod_{i \in I}\mathscr{Top}(X_i)$ of topologies on each of the sets $X_i$, let us denote by $\displaystyle\bigotimes_{i \in I}\mathscr{T}_i$ the direct product topology on the cartesian product $\displaystyle\prod_{i \in I}X_i$. If $p_i \colon \displaystyle\prod_{i \in I}X_i \to X_i$ denotes the canonical projection, then it follows straight from the definition that given an arbitrary topological space $(Y, \mathscr{S})$ and a map $f \colon Y \to \displaystyle\prod_{i \in I}X_i$ we have the equivalence:
$$f \in \mathrm{Hom}_{\mathbf{Top}}\left((Y, \mathscr{S}), \left(\displaystyle\prod_{i \in I}X_i, \displaystyle\bigotimes_{i \in I}\mathscr{T}_i\right)\right) \Leftrightarrow (\forall i)\left(i \in I \Rightarrow p_i \circ f \in \mathrm{Hom}_{\mathbf{Top}}((Y, \mathscr{S}), (X_i, \mathscr{T}_i))\right),$$
which is a very stiff and formal way of saying that $f$ is continuous if and only if all the compositions $p_i \circ f$ are continuous for every $i \in I$.
In the terminology Bourbaki is an adept of, the direct product topology is the initial structure induced by the family of spaces $\left(X_i, \mathscr{T}_i\right)_{i \in I}$ on the cartesian product $\displaystyle\prod_{i \in I}X_i$ via the family of canonical projections.
