| [ilmath]\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}[/ilmath]
|
TODO: Caption
|
Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be an arbitrary family of
topological spaces and let [ilmath](Y,\mathcal{ K })[/ilmath] be a topological space. Consider [ilmath](\prod_{\alpha\in I}X_\alpha,\mathcal{J})[/ilmath] as a topological space with
topology ([ilmath]\mathcal{J} [/ilmath]) given by the
product topology of [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath]. Lastly, let [ilmath]f:Y\rightarrow\prod_{\alpha\in I}X_\alpha[/ilmath] be a
map, and for [ilmath]\alpha\in I[/ilmath] define [ilmath]f_\alpha:Y\rightarrow X_\alpha[/ilmath] as [ilmath]f_\alpha=\pi_\alpha\circ f[/ilmath] (where [ilmath]\pi_\alpha[/ilmath] denotes the [ilmath]\alpha^\text{th} [/ilmath]
canonical projection of the product topology) then:
- [ilmath]f:Y\rightarrow\prod_{\alpha\in I}X_\alpha[/ilmath] is continuous
if and only if
- [ilmath]\forall\beta\in I[f_\beta:Y\rightarrow X_\beta\text{ is continuous}][/ilmath] - in words, each component function is continuous
TODO: Link to diagram
Proof
(Unknown grade)
This page requires one or more proofs to be filled in, it is on a
to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see
template:Caution et al).
Notes
References
