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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 sets, family of Filter (set theory), filters on ''X''. Convergence spaces generalize the notions of Filters in topology, convergence that are found in point-set topology, including Metric space, 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. An example of convergence that is in general non-topological is almost everywhere convergence. Many topological properties have generalizations to convergence spaces. Besides its ability to describe notions of convergence that Topology (structure), topologies are unable to, the Category (mathematics), category of convergence spaces has an important categorical property that the category of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows. Formal definition Let ''C'' be a category. A subcategory ''S'' of ''C'' is given by *a subcollection of objects of ''C'', denoted ob(''S''), *a subcollection of morphisms of ''C'', denoted hom(''S''). such that *for every ''X'' in ob(''S''), the identity morphism id''X'' is in hom(''S''), *for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''), *for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. These conditions ensure that ''S'' is a category in its own right: its collection of objects is ob(''S'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indiscrete Topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space can be viewed as a pseudometric space in which the distance between any two points is zero. Details The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space ''X'' with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space. Other properties of an indiscrete space ''X''—many of which are quite unusual—include: * The only closed sets are the empty set and ''X''. * The only possible basi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis. Formal definition Let ''X'' be a topological space. Most commonly ''X'' is called locally compact if every point ''x'' of ''X'' has a compact neighbourhood, i.e., there exists an open set ''U'' and a compact set ''K'', such that x\in U\subseteq K. There are other common definitions: They are all equivalent if ''X'' is a Hausdorff space (or preregular). But they are not equivalent in general: :1. every point of ''X'' has a compact neighbourhood. :2. every point of ''X'' has a closed compact neighbourhood. :2′. every point of ''X'' has a relatively compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood Filter
In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbourhood of a point or set An of a point (or subset) x in a topological space X is any open subset U of X that contains x. A is any subset N \subseteq X that contains open neighbourhood of x; explicitly, N is a neighbourhood of x in X if and only if there exists some open subset U with x \in U \subseteq N. Equivalently, a neighborhood of x is any set that contains x in its topological interior. Importantly, a "neighbourhood" does have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods." Similarly, a neighbourhood that is also a closed (respectively, compact, connected, etc.) set is called a (respectively, , , etc.). There are many other types of neighbourhoods that are used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Set
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger than ''s'' (that is, if s < x), then ''x'' is in ''S''. In other words, this means that any ''x'' element of ''X'' that is to some element of ''S'' is necessarily also an element of ''S''. The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset ''S'' of ''X'' with the property that any element ''x'' of ''X'' that is to some element of ''S'' is necessarily also an element of ''S''. Definition Let be a preordered set. An in (also called an , an ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x, y), where x is an element of X and y is an element of Y. It encodes the common concept of relation: an element x is ''related'' to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime p is related to each integer z that is a Divisibility, multiple of p, but not to an integer that is not a Multiple (mathematics), multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as -4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13. Binary relations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrafilter (set Theory)
In the mathematical field of set theory, an ultrafilter on a set (mathematics), set X is a ''maximal filter'' on the set X. In other words, it is a collection of subsets of X that satisfies the definition of a filter (set theory), filter on X and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of X that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set X can also be characterized as a filter on X with the property that for every subset A of X either A or its complement X\setminus A belongs to the ultrafilter. Ultrafilters on sets are an important special instance of Ultrafilter, ultrafilters on partially ordered sets, where the partially ordered set consists of the power set \wp(X) and the partial order is subset inclusion \,\subseteq. This article deals specifically with ultrafilters on a set and does not cover the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prefilter
In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A\subset B\subset X and A\in \mathcal, then B\in \mathcal A filter on a set may be thought of as representing a "collection of large subsets", one intuitive example being the neighborhood filter. Filters appear in order theory, model theory, and set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal. Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book '' Topologie Générale'' as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Intersection Property
In general topology, a branch of mathematics, a non-empty family A of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of A is infinite. Sets with the finite intersection property are also called centered systems and filter subbases. The finite intersection property can be used to reformulate topological compactness in terms of closed sets; this is its most prominent application. Other applications include proving that certain perfect sets are uncountable, and the construction of ultrafilters. Definition Let X be a set and \mathcal a nonempty family of subsets of that is, \mathcal is a nonempty subset of the power set of Then \mathcal is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
