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November 5th, 2010, 08:25 AM   #1
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continuous map

Let f : (X x Y) --> R be continuous where Y is compact. Show
that the map g : X --> R defined as g(x) = sup{f(x, y), taken over y, is
continuous.
Any help would be appreciated.
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November 5th, 2010, 09:48 AM   #2
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Re: continuous map

is well-defined since is compact.
, which is open.
If then For every there is an open neighborhood of in ,
and an open neighborhood of such that whenever Since is compact, there are
such that Let Then is open, and so on
Furthermore, implying that

Maybe thereīs a shorter argument, but I donīt see it.
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November 5th, 2010, 01:46 PM   #3
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Re: continuous map

Perhaps I should mention that X, Y are locally compact and Hausdorff. Regardless, thanks for the help; your proof looks correct.
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