# Continuity of the projection function

I have the following definition of the projection on product topology:

Let $$\{X_i\}_{I\in I}$$ be a family of topological spaces and for $$j\in I$$ let $$p_j:\prod\limits_{I\in I}X_i\rightarrow X_j$$ be the projection onto the $$j$$th factor i.e. $$p_j((x_i)_{I\in I})=x_j$$. Then

• For any $$j\in I$$ the function $$p_j$$ is continuous.

• A function $$f:Y\rightarrow\prod\limits_{i\in I}X_i$$ is continuous if and only if the composition $$p_j\circ f:Y\rightarrow X_j$$ is continuous $$\forall j\in I$$.

However, I'm looking for a proof of this. Everything I've looked at gave part 1. as defined in product topology, but I have yet to see a proof of it (I imagine part 2. follows).

If anyone could provide a proof, that would be great.

Take any open set $U_j\subset X_j$ then $p_j^{-1}(U_j)=X_1\times X_2\times \ldots X_{j-1}\times U_j\times X_{j+1}\times\ldots X_n\times \ldots$ which is open in $\prod_{j\in \Bbb N} X_j$
• Why is this open? Is it because each $X_i$ is also open? Oct 26, 2017 at 2:41
• Yes any open set in $\prod X_j$ is of the form $U_1\times Y_2\times \ldots U_n\times X_{n+1}\times \ldots$ Oct 26, 2017 at 5:25
• i.e finitely many are open in $X_i$ and the rest are the whole spaces Oct 26, 2017 at 5:26