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?



How to understand the concept of compact space
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. 


