In mathematics, especially analysis, exhaustion by compact sets of an open set E in the Euclidean space Rⁿ (or a manifold with countable base) is an increasing sequence of compact sets K_j, where by increasing we mean K_j is a subset of K_{j + 1}, with the limit (union) of the sequence being E. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... 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... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... 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. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

Sometimes one requires the sequence of compact sets to satisfy one more property— that K_j is contained in the interior of K_{j + 1} for each j. This, however, is dispensed in Rⁿ or a manifold with countable base. In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

For example, consider a unit open disk and the concentric closed disk of each radius inside. That is let E = {z; | z | < 1} and . Then taking the limit (union) of the sequence K_j gives E. The example can be easily generalized in other dimensions.

See also: Sigma-compact. In topology, a &#963;-compact space is a topological space that is the union of countably many compact subsets. ...

References

• Leon Ehrenpreis, Theory of Distributions for Locally Compact Spaces, American Mathematical Society, 1982. ISBN 0-8218-1221-1.

External links

• Exhaustion by compact sets on PlanetMath