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Scott Continuity
In mathematics, given two partially ordered sets ''P'' and ''Q'', a function ''f'': ''P'' → ''Q'' between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset ''D'' of ''P'' with supremum in ''P'', its image has a supremum in ''Q'', and that supremum is the image of the supremum of ''D'', i.e. \sqcup f = f(\sqcup D), where \sqcup is the directed join. When Q is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets. A subset ''O'' of a partially ordered set ''P'' is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets ''D'' with supremum in ''O'' have non-empty intersection with ''O''. The Scott-open subsets of a partially ordered set ''P'' form a topology on ''P'', the Scott topology. A functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Denotational Semantics
In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach of formalizing the meanings of programming languages by constructing mathematical objects (called ''denotations'') that describe the meanings of Expression (computer science), expressions from the languages. Other approaches providing formal semantics of programming languages include axiomatic semantics and operational semantics. Broadly speaking, denotational semantics is concerned with finding mathematical objects called domain theory, domains that represent what programs do. For example, programs (or program phrases) might be represented by partial functionsDana S. ScottOutline of a mathematical theory of computation Technical Monograph PRG-2, Oxford University Computing Laboratory, Oxford, England, November 1970.Dana Scott and Christopher Strachey. ''Toward a mathematical semantics for computer languages'' Oxford Programming Research Group Techn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lattice (order)
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join (mathematics), join) and a unique infimum (also called a greatest lower bound or meet (mathematics), meet). An example is given by the power set of a set, partially ordered by Subset, inclusion, for which the supremum is the Union (set theory), union and the infimum is the Intersection (set theory), intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic Identity (mathematics), identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilatti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Sober Space
In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible space, irreducible closed subset of ''X'' is the closure (topology), closure of exactly one point of ''X'': that is, every nonempty irreducible closed subset has a unique generic point. Definitions Sober spaces have a variety of cryptomorphic definitions, which are documented in this section. In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the Kolmogorov space, T0 axiom. Replacing it with "at least one" is equivalent to the property that the T0 quotient space (topology), quotient of the space is sober, which is sometimes referred to as having "enough points" in the literature. With irreducible closed sets A closed set is Hyperconnected space, irreducible if it cannot be written as the union of two proper closed subsets. A space is sober if every nonempty irreducible closed subset is the closure of a unique point. In terms of morphisms o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Open Neighbourhood
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set. Definitions Neighbourhood of a point If X is a topological space and p is a point in X, then a neighbourhood of p is a subset V of X that includes an open set U containing p, p \in U \subseteq V \subseteq X. This is equivalent to the point p \in X belonging to the topological interior of V in X. The neighbourhood V need not be an open subset of X. When V is open (resp. closed, compact, etc.) in X, it is called an (resp. closed neighbourhood, compact neighbourhood, etc.). Some authors require neighbourhoods to be open, so it is important to note their conventions. A set that is a neighbourhood ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Specialization Order
In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the T0 separation axiom, this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little interest. The specialization order is often considered in applications in computer science, where T0 spaces occur in denotational semantics. The specialization order is also important for identifying suitable topologies on partially ordered sets, as is done in order theory. Definition and motivation Consider any topological space ''X''. The specialization preorder ≤ on ''X'' relates two points of ''X'' when one lies in the closure of the other. However, various authors disagree on which 'direction' the order should go. What is agreed is that if :''x'' is containe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
