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Stone Space
In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in the 1930s in the course of his investigation of Boolean algebras, which culminated in his representation theorem for Boolean algebras. Equivalent conditions The following conditions on the topological space X are equivalent: * X is a Stone space; * X is homeomorphic to the projective limit (in the category of topological spaces) of an inverse system of finite discrete spaces; * X is compact and totally separated; * X is compact, T0 , and zero-dimensional (in the sense of the small inductive dimension); * X is coherent and Hausdorff. Examples Important examples of Stone spaces include finite discrete spaces, the Cantor set and the space \Z_p of p-adic integers, where p is any prime number. Generalizing these examples, any prod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a '' topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Definitions Given a set X: A metric space (E,d) is said to be '' uniformly discrete'' if there exists a ' r > 0 such that, for any x,y \in E, one has either x = y or d(x,y) > r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set \left\. Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Of Categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation. If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent. An equivalence of categories consists of a functor between the involved categories, which is required ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone Duality
In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics. This article gives pointers to special cases of Stone duality and explains a very general instance thereof in detail. Overview of Stone-type dualities Probably the most general duality that is classically referred to as "Stone duality" is the duality between the category Sob of sober spaces with continuous functions and the category SFrm of spatial frames with appropriate frame homomorphisms. The dual category of SFrm is the category of spati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, (C^)^ = C. Examples * An example comes from reversing the direction of inequalities in a partial order. So if ''X'' is a set and ≤ a partial order relation, we can define a new partial order relation ≤op by :: ''x'' ≤op ''y'' if and only if ''y'' ≤ ''x''. : The new order is commonly called dual order of ≤, and is mostly denoted by ≥. Therefore, duality plays an important role in order theory and every purely order theoretic concept has a dual. For example, there are opposite pairs child/parent, descendant/ancestor, infimum/ supremum, down-set/ up-set, ideal/filter etc. This order theoretic duality is in turn a spe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a linguistic context; see function word. Definition Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y) in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clopen Set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for is unrelated to their meaning for (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as " door spaces" their name. Examples In any topological space X, the empty set and the whole space X are both clopen. Now consider the space X which consists of the union of the two op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P. If X is an arbitrary set, its power set \wp(X), ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on \wp(X) are usually called X.If X happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on \wp(X) or an (ultra)filter just on X is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter" ''of'' a partial ordered set" vs. "''on'' an arbitrary set"; i.e. they write "(ultra)filter on X" to abbreviate "(ultra)filter of \wp(X)". An ultrafilter on a set X may be considered as a finitely additive measure on X. In this view, every subset of X is either considered " almost everything" (has measure 1) or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–Čech Compactification
In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Stone–Čech compactification ''βX'' of a topological space ''X'' is the largest, most general compact Hausdorff space "generated" by ''X'', in the sense that any continuous map from ''X'' to a compact Hausdorff space factors through ''βX'' (in a unique way). If ''X'' is a Tychonoff space then the map from ''X'' to its image in ''βX'' is a homeomorphism, so ''X'' can be thought of as a (dense) subspace of ''βX''; every other compact Hausdorff space that densely contains ''X'' is a quotient of ''βX''. For general topological spaces ''X'', the map from ''X'' to ''βX'' need not be injective. A form of the axiom of choice is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Profinite Group
In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists d\in\N such that every group in the system can be generated by d elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems. To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the profi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product. Definition Throughout, I will be some non-empty index set and for every index i \in I, let X_i be a topological space. Denote the Cartesian product of the sets X_i by X := \prod X_ := \prod_ X_i and for every index i \in I, denote the i-th by \begin p_i :\;&& \prod_ X_j &&\;\to\; & X_i \\ .3ex && \l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
