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Can any one tell me in simple words what is a compact set? I read the definition of Compact set, but do not get it. BTW, I do not know topology.

In particular, is the probability simplex, $W\ge0, W1=1$, a compact set?

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Try this one. –  HipsterMathematician Dec 3 '12 at 20:23
    
A set is compact if it is closed and bounded. –  HipsterMathematician Dec 3 '12 at 20:24
    
Closed and bounded, you mean. But that's only true in locally compact spaces, e.g. ${\mathbb R}^n$. –  Robert Israel Dec 3 '12 at 20:26
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The problem with that intuition, @Neal, is that the subset of a compact set is not necessarily compact, which violates our intuition of "small." –  Thomas Andrews Dec 3 '12 at 20:32
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And yes, the $n$-simplex, whether the probability simplex or any other definition of the same topology, is a compact space. –  Thomas Andrews Dec 3 '12 at 20:37

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up vote 4 down vote accepted

One useful characterization, especially as regards optimization: a set $S$ (in a metric space) is compact if and only if every continuous function on $S$ has a maximum.

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Presumably, you mean continuous real-valued function... –  Thomas Andrews Dec 3 '12 at 20:34
    
Yes, that's what I mean. –  Robert Israel Dec 3 '12 at 20:42

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