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Nov 30, 2012 at 10:35 comment added Pietro Majer Indeed, I am saying that countably compact is a simple sufficient condition for all real valued continuous function have maximum. The example of a pseudocompact, non countably compact space $X$ should be in "counterexamples in topology".
Nov 30, 2012 at 9:40 comment added djoke @Pietro Majer. See the email. The metric space has this property, and need not be a countably compact space.
Nov 30, 2012 at 7:46 history answered Pietro Majer CC BY-SA 3.0