Convergence space

In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a that satisfies certain properties relating elements of X with the family of filters on X. Convergence spaces generalize the notions of convergence that are found in point-set topology, including metric convergence and uniform convergence. Every topological space gives rise to a canonical convergence but there are convergences, known as , that do not arise from any topological space. Examples of convergences that are in general non-topological include convergence in measure and almost everywhere convergence. Many topological properties have generalizations to convergence spaces. Besides its ability to describe notions of convergence that topologies are unable to, the of convergence spaces has an important categorical property that the lacks. The category of topological spaces is not an exponential category (or equivalently, it is not Cartesian closed) although it is contained in the exponential category of pseudotopological spaces, which is itself a subcategory of the (also exponential) category of convergence spaces. Filters in topology and Ultrafilter Denote the power set of a set by The or in of a family of subsets is defined as and similarly the of is If (resp. ) then is said to be (resp. ) in For any families and declare that if and only if for every there exists some such that or equivalently, if then if and only if The relation defines a preorder on If which by definition means then is said to be and also and is said to be The relation is called . Two families and are called ( ) if and A is a non-empty subset that is upward closed in closed under finite intersections, and does not have the empty set as an element (i.e. ). A is any family of sets that is equivalent (with respect to subordination) to filter or equivalently, it is any family of sets whose upward closure is a filter.