Characteristic property of the product topology
From Maths
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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
Contents
Statement
 [ilmath]f:Y\rightarrow\prod_{\alpha\in I}X_\alpha[/ilmath] is continuous
 [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
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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)
Notes
References
