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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](http://en.wikipedia.org/wiki/Compact_set).2012-12-03
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    A set is compact if it is closed and bounded.2012-12-03
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    Closed and bounded, you mean. But that's only true in locally compact spaces, e.g. ${\mathbb R}^n$.2012-12-03
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    @Charlie is right in most cases that you'll care about at the start. But if you don't know what "closed" is (it is a topological term) it will probably still need further explanation.2012-12-03
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    @RobertIsrael ,@ThomasAndrews Thanks a lot!2012-12-03
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    Compact sets are "small", the next step up from finite.2012-12-03
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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."2012-12-03
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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.2012-12-03
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    Closed and *totally* bounded.2012-12-14

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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...2012-12-03
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    Yes, that's what I mean.2012-12-03