the definition of compact space is: A subset K of a metric space X is said to be compact if every open cover of K contains finite subcovers. What is the meaning of defining a space is "compact". I found the explanation on wikipedia : "In mathematics, more specifically general topology and metric topology, a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space. " I can not understand it, or at least I can not get this by its definition. Can anybody help me?
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3$\begingroup$ Well suppose you have compact metric space $X$, consider for any $\varepsilon>0$ the covering $\{B_\varepsilon(x)|x\in X\}$. It has a finite subcovering. Now take any infinite sequence $(x_i)_{i \in \mathbb{N}}$. Each $x_i$ is contained in one open set of this finite subcovering. Hence (for fixed $\varepsilon>0$) there must be a point $x$ such that infinitely many $x_i$ are contained in $B_\varepsilon(x)$. $\endgroup$– HenrikRüpingCommented May 26, 2010 at 8:51
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$\begingroup$ The intuition of the idea comes from metric space theory: suppose you took an 'infinite number of steps' to get from point A to point B, then each step could be covered by a ball around the centre of the step. The worst you could have done to take an infinite number of steps would have been to 'traverse the whole space' (so your balls cover the space), but then compactness would guarantee that a finite number of them would have done the job anyway. Very roughly. Really what you're after is the equivalence of sequential and covering compactness. See: any metric space textbook ever. $\endgroup$– Tom BoardmanCommented May 26, 2010 at 9:00
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$\begingroup$ How the standard definition of compactness came about is a bit mysterious to me as well, although perhaps less so than the definition of a topology itself: mathoverflow.net/questions/19152/… $\endgroup$– Minhyong KimCommented May 26, 2010 at 23:00
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$\begingroup$ Closed; mathoverflow is for questions of interest to research mathematicians. $\endgroup$– Kim MorrisonCommented May 26, 2010 at 23:50
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6 Answers
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Some heuristic remarks are helpful only to a subset of readers. (Maybe that's true of all heuristics, as a meta-heuristic - if everyone accepts a rough explanation, it's something rather more than that.)
Non-compactness is about being able to "move off to infinity" in some way in a space. On the real line you can do that to the left, or right: but bend the line round to fill all but one point on a circle (which is compact) and you see the difference having the "other point" near which you end up. This example of real line versus circle is too simple, really. Another way you can "go off to infinity" in a space is by having paths branching out infinitely (as in König's lemma, which supplies another kind of intuition).
Compactness is a major topological concept because the various ways you might try to "trap" movement within a space to prevent "escape" to infinity can be summed up in a single idea (for metric spaces, let's say). The definition by open sets is cleaner, but the definition by sequences having to accumulate on themselves (not necessarily to converge, but to have at least one convergent subsequence) is somewhat quicker to say. If you restrict attention to spaces that are manifolds, you can think of continuous paths and whether they have to wind back close to themselves or not.
answered May 26, 2010 at 9:32
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$\begingroup$ Your answer gives me a very intuitive view to compactness! Thanks very much! $\endgroup$– jkjiumCommented May 26, 2010 at 10:07
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Terry Tao has a nice explanation in the Princeton companion to maths. The article's also on his website:
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2$\begingroup$ +1. I have been searching, for a while, for a good motivation of compactness. I read the article by Terry Tao and it is really illuminating. Thanks for the link! $\endgroup$– user11000Commented Aug 16, 2011 at 17:03
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Etymology may help. Compact, from the Latin Compactus, past participle of the verb Compingere: "to pack together closely and firmly". You may have an idea of how strong is the hypothesis of compactness if you look at what happens when even total boundedness is lacking.
Just consider e.g. the possibly most familiar infinite dimensional object, the separable Hilbert space H=l2. Its unit closed ball is not compact. A continuous real valued function on it may be unbounded, or bounded without minimum and maximum value. A linear form on H may be everywhere discontinuous and locally unbounded. There is a continuous transformation of the unit closed ball with no fixed points. The ball itself retracts on its boundary. Infinitely many disjoint unit open balls may be packed within a ball of radius $1+ \sqrt{2}$. The linear group GL(H) is connected. There are injective linear continuous transformations of H that are not surjective....
answered May 26, 2010 at 10:49
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In elementary analysis one learns that every continuous function from a closed bounded interval to $\mathbb{R}$ is bounded, but this is not the case for open or unbounded intervals. A little later one learns that each continuous function from a closed bounded subset of $\mathbb{R}^n$ to $\mathbb{R}$ is bounded, but this fails for other subsets of $\mathbb{R}^n$. The key properties are that a subset of $\mathbb{R}^n$ is compact iff it is closed and bounded (Heine-Borel theorem) and that the image of a compact space under a continuous map is compact. Compactness is a sufficient condition on a space to ensure that all continuous functions to $\mathbb{R}$; moreover compactness is a purely topological property, definable in terms of open sets.
The wikipedia quotation is a bit vague, but it refers to a property called sequential compactness, which all compact metric spaces have. This means that for any sequence of points in the space $X$ there is a subsequence converging to a point of $X$. A metric space is compact if and only if it is sequentially compact, but this does not hold for non-metrizable topological spaces.
I don't like the wikipedia quote, as it sort of suggests that sequences in compact spaces must be convergent; this is not so, though they must have convergent subsequences.
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A very nice exposition was given by Hewitt in this article in the Monthly; the central thesis is that compactness facilitates proving results that are nigh on trivial for finite sets in wider settings.
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$\begingroup$ Yep. I've heard people informally describe a compact set as one that is "almost finite". $\endgroup$ Commented May 26, 2010 at 15:15
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In some way, you can ALWAYS think about compactness as "sequential compactness" provided you allow for things a bit more general than sequences- nets. Compactness is equivalent to the statement that every net as a convergent subnet. If a space is metrizable, this is equivalent to the analogous statement, where you only need to consider sequences. I prefer, in fact, to think in terms of ultrafilters and say that a space is compact if and only if every ultrafilter has a limit point. I suggest reading up about nets and ultrafilters. I found http://www.math.ksu.edu/~nagy/real-an/1-02-convergence.pdf a nice quick introduction to give you the idea (but it does not classify compactness). A more comprehensive reference might be http://alpha.math.uga.edu/~pete/convergence.pdf (but note here compact means compact Hausdorff, while quasi-compact means compact and possibly not Hausdorff).
