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Upper Topology
In mathematics, the upper topology on a partially ordered set ''X'' is the coarsest topology in which the Closure (topology), closure of a singleton (mathematics), singleton \ is the order section a] = \ for each a\in X. If \leq is a partial order, the upper topology is the least Specialization_(pre)order#Important_properties, order consistent topology in which all open sets are up-sets. However, not all up-sets must necessarily be open sets. The lower topology induced by the preorder is defined similarly in terms of the down-sets. The preorder inducing the upper topology is its specialization preorder, but the specialization preorder of the lower topology is opposite to the inducing preorder. The real upper topology is most naturally defined on the upper-extended real line (-\infty, +\infty] = \R \cup \ by the system \ of open sets. Similarly, the real lower topology \ is naturally defined on the lower real line [-\infty, +\infty) = \R \cup \. A real function on a 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] |
Partially Ordered Set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is Reflexive relation, reflexive, antisymmetric relation, antisymmetric, and Transitive relation, transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coarsest Topology
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the collection of subsets which are considered to be "open". (An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.) For definiteness the reader should think of a topology as the family of open sets of a topological space, since that is the standard meaning of the word "topology". Let ''τ''1 and ''τ''2 be two topologies on a set ''X'' such that ''τ''1 is contained in ''τ''2: :\tau_1 \subseteq \tau_2. That is, every element of ''τ''1 is also an element of ''τ''2. Then the topology ''τ''1 is said to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (topology)
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of all closed sets containing . Intuitively, the closure can be thought of as all the points that are either in or "very near" . A point which is in the closure of is a point of closure of . The notion of closure is in many ways dual to the notion of interior. Definitions Point of closure For S as a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point can be x itself). This definition generalizes to any subset S of a metric space X. Fully expressed, for X as a metric space with metric d, x is a point of closure of S if for every r > 0 there exists some s \in S such that the distance d(x, s) < r ( is allowed). Another way to expre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singleton (mathematics)
In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and \ are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as \ is a singleton as it contains a single element (which itself is a set, but not a singleton). A set is a singleton if and only if its cardinality is . In von Neumann's set-theoretic construction of the natural numbers, the number 1 is ''defined'' as the singleton \. In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set ''A'', the axiom applied to ''A'' and ''A'' asserts the existence of \, which is the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Specialization (pre)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 conta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, an open set is a member of a given Set (mathematics), collection of Subset, subsets of a given set, a collection that has the property of containing every union (set theory), union of its members, every finite intersection (set theory), intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology (structure), topology. These conditions are very loose, and allow enormous flexibility in the choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up-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 . An in[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Specialization Preorder
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 contain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper-extended Real Line
In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities. For example, the infinite sequence (1,2,\ldots) of the natural numbers increases ''infinitively'' and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has +\infty as its least upper bound and as its limit (an actual infinity). In calculus and mathematical analysis, the use of +\infty and -\infty as actual limits extends significantly the possible computations. It is the Dedekind–MacNeille completion of the real numbers. The extended real number system is denoted \overline, \infty,+\infty/math>, or \R\cup\left\. When the meaning is clear from context, the symbol +\infty is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
