November 5th, 2010, 08:25 AM  #1 
Newbie Joined: Jan 2009 Posts: 20 Thanks: 0  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. 
November 5th, 2010, 09:48 AM  #2 
Newbie Joined: Oct 2010 Posts: 17 Thanks: 0  Re: continuous map is welldefined 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. 
November 5th, 2010, 01:46 PM  #3 
Newbie Joined: Jan 2009 Posts: 20 Thanks: 0  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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