Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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
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
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
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.