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. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X...

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 Bolzano-Weierstrass theorem, which would say that if a and b are numbers such that a < b, then the set [a, b] is compact. This is a page about mathematics. ... Please refer to Real vs. ... In mathematics, a subsequence of some sequence is a new sequence which is formed from the original sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. ... The Bolzano-Weierstrass theorem in real analysis states that every bounded sequence of real numbers contains a convergent subsequence. ...

Heine-Borel theorem: In Rⁿ a set is compact if and only if it is closed and bounded. In mathematical analysis, the Heine-Borel theorem states: A subset of the real numbers R is compact iff it is closed and bounded. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ...

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. In the mathematical area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any directed set that does not already contain members above the compact element. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... See lattice for other mathematical as well as non-mathematical meanings of the term. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

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. The word Boundary has a variety of meanings. ...

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. Non-tangency 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. In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... See also Asymptotic analysis but contrast asymptotic curve. ...

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 P₁ in the given set at a distance less than ε from the chosen boundary point. Then, due to the requirement of non-tangency, pick a ball inside the open set which is not tangent to the boundary. Call it C₁. C₁ will cover P₁, but then one can pick another point P₂ even closer to B than P₁ but which is not in C₁. Then pick an open ball C₂ whose boundary is not tangent to the boundary of the given set, but which includes P₂... This process can go on for P₃, C₃, P₄, C₄, P₅, C₅, 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. A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...

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. In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ...

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 . 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. In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...