November 5th, 2010, 09:25 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, 10:48 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, 02:46 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.


Tags 
continuous, map 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
continuous for 1/x  frankpupu  Calculus  2  February 9th, 2012 06:43 PM 
continuous  frankpupu  Calculus  4  January 31st, 2012 06:17 AM 
Proving that a space is continuous, continuous at 0, and bdd  thedoctor818  Real Analysis  17  November 9th, 2010 09:19 AM 
continuous at 0  summerset353  Real Analysis  2  February 22nd, 2010 04:37 PM 
continuous at any point iff continuous at origin  babyRudin  Real Analysis  6  October 24th, 2008 01:58 AM 