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Corona Set
In mathematics, the corona or corona set of a topological space ''X'' is the complement β''X''\''X'' of the space in its Stone–Čech compactification β''X''. A topological space is said to be σ-compact if it is the union of countably many compact subspaces, and locally compact if every point has a neighbourhood with compact closure. The corona of a σ-compact and locally compact Hausdorff space is a sub-Stonean space, i.e., any two open σ-compact disjoint subsets have disjoint compact closures. See also * Corona theorem *Corona algebra, a non-commutative analogue of the corona set. References *{{Citation , last1=Grove , first1=Karsten , last2=Pedersen , first2=Gert Kjærgård , title=Sub-Stonean spaces and corona sets , doi=10.1016/0022-1236(84)90028-4 , mr=735707 , year=1984 , journal=Journal of Functional Analysis The ''Journal of Functional Analysis'' is a mathematics journal published by Elsevier. Founded by Paul Malliavin, Ralph S. Phillips, and Ir ... [...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 ... [...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 "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 express this is to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corona Algebra
In mathematics, the multiplier algebra, denoted by ''M''(''A''), of a C*-algebra ''A'' is a unital C*-algebra that is the largest unital C*-algebra that contains ''A'' as an ideal in a "non-degenerate" way. It is the noncommutative generalization of Stone–Čech compactification. Multiplier algebras were introduced by . For example, if ''A'' is the C*-algebra of compact operators on a separable Hilbert space, ''M''(''A'') is ''B''(''H''), the C*-algebra of all bounded operators on ''H''. Definition An ideal ''I'' in a C*-algebra ''B'' is said to be essential if ''I'' ∩ ''J'' is non-trivial for every ideal ''J''. An ideal ''I'' is essential if and only if ''I''⊥, the "orthogonal complement" of ''I'' in the Hilbert C*-module ''B'' is . Let ''A'' be a C*-algebra. Its multiplier algebra ''M''(''A'') is any C*-algebra satisfying the following universal property: for all C*-algebra ''D'' containing ''A'' as an ideal, there exists a unique *-homomorphism φ: ''D'' → ''M''(' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corona Theorem
In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by and proved by . The commutative Banach algebra and Hardy space ''H''∞ consists of the bounded holomorphic functions on the open unit disc ''D''. Its spectrum ''S'' (the closed maximal ideals) contains ''D'' as an open subspace because for each ''z'' in ''D'' there is a maximal ideal consisting of functions ''f'' with :''f''(''z'') = 0. The subspace ''D'' cannot make up the entire spectrum ''S'', essentially because the spectrum is a compact space and ''D'' is not. The complement of the closure of ''D'' in ''S'' was called the corona by , and the corona theorem states that the corona is empty, or in other words the open unit disc ''D'' is dense in the spectrum. A more elementary formulation is that elements ''f''1,...,''f''''n'' generate the unit ideal of ''H''∞ if and only if there is some δ>0 such that :, f_1, +\cdots+, f_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Sets
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. Generalizations This definition of disjoint sets can be extended to a family of sets \left(A_i\right)_: the family is pairwise disjoint, or mutually disjoint if A_i \cap A_j = \varnothing whenever i \neq j. Alternatively, some authors use the term disjoint to refer to this notion as well. For families the notion of pairwise disjoint or mutually disjoint is sometimes defined in a subtly different manner, in that repeated identical members are allowed: the family is pairwise disjoint if A_i \cap A_j = \varnothing whenever A_i \neq A_j (every two ''distinct'' sets in the family are disjoint).. For example, the collection of sets is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite 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. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice, however, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sub-Stonean Space
In topology, a sub-Stonean space is a locally compact Hausdorff space such that any two open σ-compact disjoint subsets have disjoint compact closures. Related, an F-space, introduced by , is a completely regular Hausdorff space for which every finitely generated ideal of the ring of real-valued continuous functions is principal, or equivalently every real-valued continuous function f can be written as f=g, f, for some real-valued continuous function g. When dealing with compact spaces, the two concepts are the same, but in general, the concepts are different. The relationship between the sub-Stonean spaces and F-spaces is studied in Henriksen and Woods, 1989. Examples Rickart spaces and the corona sets of locally compact σ-compact Hausdorff spaces are sub-Stonean spaces. See also * Extremally disconnected space * F-space References * * *{{Citation , last1=Henriksen, first1=Melvin, last2=Woods, first2=R. G., title=F-Spaces and Substonean Spaces: General Topology as a Tool in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighbourhood (mathematics)
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 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 also equivalent to the point p \in X belonging to the topological interior of V in X. The neighbourhood V need be an open subset X, but when V is open in X then it is called an . Some authors have been known to require neighbourhoods to be open, so it is important to note conventions. A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some 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 mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact Space
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. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. 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 compact neighbourho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''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 topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
