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Pairwise Stone Space
In mathematics and particularly in topology, pairwise Stone space is a bitopological space \scriptstyle (X,\tau_1,\tau_2) which is pairwise compact, pairwise Hausdorff, and pairwise zero-dimensional. Pairwise Stone spaces are a bitopological version of the Stone spaces. Pairwise Stone spaces are closely related to spectral spaces. Theorem:G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). Bitopological duality for distributive lattices and Heyting algebras. ''Mathematical Structures in Computer Science'', 20. If \scriptstyle (X,\tau) is a spectral space, then \scriptstyle (X,\tau,\tau^*) is a pairwise Stone space, where \scriptstyle \tau^* is the de Groot dual topology of \scriptstyle \tau . Conversely, if \scriptstyle (X,\tau_1,\tau_2) is a pairwise Stone space, then both \scriptstyle (X,\tau_1) and \scriptstyle (X,\tau_2) are spectral spaces. See also * Bitopological space * Duality theory for distributive lattices In mathematics, duality theory f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...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] |
Bitopological Space
In mathematics, a bitopological space is a set endowed with ''two'' topologies. Typically, if the set is X and the topologies are \sigma and \tau then the bitopological space is referred to as (X,\sigma,\tau). The notion was introduced by J. C. Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric. Continuity A map \scriptstyle f:X\to X' from a bitopological space \scriptstyle (X,\tau_1,\tau_2) to another bitopological space \scriptstyle (X',\tau_1',\tau_2') is called continuous or sometimes pairwise continuous if \scriptstyle f is continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ... both as a map from \scriptstyle (X,\tau_1) to \scriptstyle (X',\tau_1') and as map from \scriptstyle (X,\tau_2) to \scriptstyle (X',\tau_2'). Bitop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Spectral Space
In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos. Definition Let ''X'' be a topological space and let ''K''\circ(''X'') be the set of all compact open subsets of ''X''. Then ''X'' is said to be ''spectral'' if it satisfies all of the following conditions: *''X'' is compact and T0. * ''K''\circ(''X'') is a basis of open subsets of ''X''. * ''K''\circ(''X'') is closed under finite intersections. * ''X'' is sober, i.e., every nonempty irreducible closed subset of ''X'' has a (necessarily unique) generic point. Equivalent descriptions Let ''X'' be a topological space. Each of the following properties are equivalent to the property of ''X'' being spectral: #''X'' is homeomorphic to a projective limit of finite T0-spaces. #''X'' is homeomorphic to the spectrum of a bounded distributive lattice ''L''. In this case, ''L' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Groot Dual
In mathematics, in particular in topology, the de Groot dual (after Johannes de Groot) of a topology ''τ'' on a set ''X'' is the topology ''τ''* whose closed sets are generated by compact saturated subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...s of (''X'', ''τ''). References * R. Kopperman (1995), Asymmetry and duality in topology. ''Topology Applications'', 66(1), 1–39, 1995. Topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitopological Space
In mathematics, a bitopological space is a set endowed with ''two'' topologies. Typically, if the set is X and the topologies are \sigma and \tau then the bitopological space is referred to as (X,\sigma,\tau). The notion was introduced by J. C. Kelly in the study of quasimetrics, i.e. distance functions that are not required to be symmetric. Continuity A map \scriptstyle f:X\to X' from a bitopological space \scriptstyle (X,\tau_1,\tau_2) to another bitopological space \scriptstyle (X',\tau_1',\tau_2') is called continuous or sometimes pairwise continuous if \scriptstyle f is continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ... both as a map from \scriptstyle (X,\tau_1) to \scriptstyle (X',\tau_1') and as map from \scriptstyle (X,\tau_2) to \scriptstyle (X',\tau_2'). Bitop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality Theory For Distributive Lattices
In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone, generalizes the well-known Stone duality between Stone spaces and Boolean algebras. Let be a bounded distributive lattice, and let denote the set of prime filters of . For each , let . Then is a spectral space, where the topology on is generated by . The spectral space is called the ''prime spectrum'' of . The map is a lattice isomorphism from onto the lattice of all compact open subsets of . In fact, each spectral space is homeomorphic to the prime spectrum of some bounded distributive lattice. Similarly, if and denotes the topology generated by , then is also a spectral space. Moreover, is a pairwise Stone space. The pairwise Stone space is called the ''bitopological dual'' of . Ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
