Characteristic property of the product topology

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Munkres or Lee's topological manifolds. I'll fill it in when I'm more used to the terminology. I'm not happy with it ATM

Statement

$\begin{xy} \xymatrix{ & & \prod_{\alpha\in I}X_\alpha \ar[dd] \\ & & \\ Y \ar[uurr]^f \ar[rr]+<-0.9ex,0.15ex>|(.875){\hole} & & X_b \save (15,13)+"3,3"*+{\ldots}="udots"; (8.125,6.5)+"3,3"*+{X_a}="x1"; (-8.125,-6.5)+"3,3"*+{X_c}="x3"; (-15,-13)+"3,3"*+{\ldots}="ldots"; \ar@{->} "x1"; "1,3"; \ar@{->}_(0.65){\pi_c,\ \pi_b,\ \pi_a} "x3"; "1,3"; \ar@{->}|(.873){\hole} "x1"+<-0.9ex,0.15ex>; "3,1"; \ar@{->}_{f_c,\ f_b,\ f_a} "x3"+<-0.9ex,0.3ex>; "3,1"; \restore } \end{xy}$

TODO: Caption

Let $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$ be an arbitrary family of topological spaces and let $(Y,\mathcal{ K })$ be a topological space. Consider $(\prod_{\alpha\in I}X_\alpha,\mathcal{J})$ as a topological space with topology ($\mathcal{J}$) given by the product topology of $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$. Lastly, let $f:Y\rightarrow\prod_{\alpha\in I}X_\alpha$ be a map, and for $\alpha\in I$ define $f_\alpha:Y\rightarrow X_\alpha$ as $f_\alpha=\pi_\alpha\circ f$ (where $\pi_\alpha$ denotes the $\alpha^\text{th}$ canonical projection of the product topology) then:

• $f:Y\rightarrow\prod_{\alpha\in I}X_\alpha$ is continuous

if and only if

• $\forall\beta\in I[f_\beta:Y\rightarrow X_\beta\text{ is continuous}]$ - in words, each component function is continuous

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TODO: Link to diagram

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Proof

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Due to importance of page, this proof ought to be filled in, I've done it on paper (rough and neat, here's the neat: Media:Quick proof of char prop of product top.JPG)