Compact set
From Academic Kids

In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover.
Alternately, in the context of sequence analysis, a set S of real numbers is considered compact if every sequence in S has a subsequence that converges to a point in S. As an interesting aside, this provides a concise representation of the BolzanoWeierstrass theorem, which would say that if a and b are numbers such that a < b, then the set [a, b] is compact.
HeineBorel theorem: In R^{n} a set is compact if and only if it is closed and bounded.
Note that a set within any collection of sets can be called "compact" if it is a compact element in the partially ordered set induced by the order of subset inclusion on this collection. This usage agrees with the above definition if the collection of sets forms the complete lattice of open sets of a topology.
Discussion of the theorem
If a set is not closed, then it cannot be compact.
If a set is not closed, then it is either an open set, or it is partially open: part of its boundary is open, by which is meant that that part of the boundary does not belong to the set.
Then it is possible to come up with an infinite cover whose elements (which are all, by definition, open) are all subsets of the given open set, but whose boundaries are never tangent to the open boundary of the given set. Nontangency implies that the elements in the cover will have to approach the boundary by decreasing both their diameter and their distance to the boundary asymptotically to zero.
Therefore points infinitesimally close to the boundary of the given (open) set can only be covered by infinite subcovers of the infinite cover. Why infinite subcovers? Pick a point on the boundary of the given set, then pick a point P_{1} in the given set at a distance less than ε from the chosen boundary point. Then, due to the requirement of nontangency, pick a ball inside the open set which is not tangent to the boundary. Call it C_{1}. C_{1} will cover P_{1}, but then one can pick another point P_{2} even closer to B than P_{1} but which is not in C_{1}. Then pick an open ball C_{2} whose boundary is not tangent to the boundary of the given set, but which includes P_{2}... This process can go on for P_{3}, C_{3}, P_{4}, C_{4}, P_{5}, C_{5}, etc. without end. This means that if a cover of an open boundary point has elements which are all subsets of the given set and whose boundaries are never tangent to the boundary of the given set, then this cover can not be finite, and so any such infinite cover cannot have a finite subcover.
If a set is unbounded, then it cannot be compact.
Why? Because one can always come up with an infinite cover, whose elements have an upper finite bound to their size, i.e. the elements of the cover are not allowed to grow in size without bound.
But there is no finite cover of an unbounded set such that its elements do not grow in size without bounds: if one adds up a finite set of n numbers whose upper limit is m, then their sum can be no greater than <math> n \times m <math>. A similar case holds for unbounded sets with finite covers: the elements of the finite cover could not possibly be bound in size, otherwise the union of all the elements of the finite cover would itself be bounded, and could not cover an unbounded set.
So if an unbounded set is covered with an infinite cover whose elements have an upper finite bound to their size, then this infinite cover of the given set will have no finite subcover, because any subcover will be made up of elements of the cover, but the elements of the cover have an upper bound to their size, so the elements of the subcover will also have an upper bound to their size, and there is no finite cover of an unbounded set such that its elements have an upper bound to their diameters.
Therefore, given an unbounded set, there exists at least one infinite cover which has no finite subcover: namely, an infinite cover whose elements have diameters all of which have the same finite upper bound. Thus, an unbounded set cannot be compact.
See also: compact space.zh:紧集