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Nowhere Dense Set
In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. For example, the integers are nowhere dense among the reals, whereas the interval (0, 1) is not nowhere dense. A countable union of nowhere dense sets is called a meagre set. Meagre sets play an important role in the formulation of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis. Definition Density nowhere can be characterized in different (but equivalent) ways. The simplest definition is the one from density: A subset S of a topological space X is said to be ''dense'' in another set U if the intersection S \cap U is a dense subset of U. S is or in X if S is not dense in any nonempty open subset U of X. Expanding out the negation of density, it is equivalent that ea ... [...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] |
Exterior (topology)
In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the closure of the complement of . In this sense interior and closure are dual notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty). The interior and exterior of a closed curve are a slightly different concept; see the Jordan curve theorem. Definitions Interior point If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is completely contained in S. (This is illustrated in the introductory section to this article.) This definition g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Vector Space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces. Many topological vector spaces are spaces of functions, or linear operators ac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Subspace
Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics and physics * Vector (mathematics and physics) ** Row and column vectors, single row or column matrices ** Vector quantity ** Vector space ** Vector field, a vector for each point Molecular biology * Vector (molecular biology), a DNA molecule used as a vehicle to artificially carry foreign genetic material into another cell ** Cloning vector, a small piece of DNA into which a foreign DNA fragment can be inserted for cloning purposes ** Shuttle vector, a plasmid constructed so that it can propagate in two different host species ** Viral vector, a tool commonly used by molecular biologists to deliver genetic materials into cells Computer science * Vector, a one-dimensional array data structure ** Distance-vector routing protocol, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isolated Point
In mathematics, a point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a neighborhood of that does not contain any other points of . This is equivalent to saying that the singleton is an open set in the topological space (considered as a subspace of ). Another equivalent formulation is: an element of is an isolated point of if and only if it is not a limit point of . If the space is a metric space, for example a Euclidean space, then an element of is an isolated point of if there exists an open ball around that contains only finitely many elements of . A point set that is made up only of isolated points is called a discrete set or discrete point set (see also discrete space). Related notions Any discrete subset of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of may be mapped injective ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T1 Space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms. Definitions Let ''X'' be a topological space and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' are if each lies in a neighbourhood that does not contain the other point. * ''X'' is called a T1 space if any two distinct points in ''X'' are separated. * ''X'' is called an R0 space if any two topologically distinguishable points in ''X'' are separated. A T1 space is also called an accessible space or a space with Fréchet topology and an R0 space is also called a symmetric space. (The term also has an entirely different meaning in functional analysis. For this reason, the term ''T1 space'' is preferred. There is also a n ... [...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 on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense. More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturally homeomorphic to the countable product ^ of the discrete two-point space \underline 2 . By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the Closure (topology), closure of not belonging to the Interior (topology), interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include \operatorname(S), \operatorname(S), and \partial S. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a Manifold#Manifold with boundary, different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to Felix Hausdorff, Hausdorff's border, which is defined as the intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meager Set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms. The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre. Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis. Definitions Throughout, X will be a topological space. The definition of meagre set uses the notion of a nowhere dense subset of X, that is, a subset of X whose closure has empty interior. See the corresponding article for more details. A subset of X is called X, a of X, or o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma-ideal
In mathematics, particularly measure theory, a -ideal, or sigma ideal, of a σ-algebra (, read "sigma") is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is in probability theory. Let (X, \Sigma) be a measurable space (meaning \Sigma is a -algebra of subsets of X). A subset N of \Sigma is a -ideal if the following properties are satisfied: # \varnothing \in N; # When A \in N and B \in \Sigma then B \subseteq A implies B \in N; # If \left\_ \subseteq N then \bigcup_ A_n \in N. Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. The concept of -ideal is dual to that of a countably complete (-) filter. If a measure \mu is given on (X, \Sigma), the set of \mu- negligible sets (S \in \Sigma such that \mu(S) = 0) is a -ideal. The notion can be generalized to preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negligible Set
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function. Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere. In order for these to work, it is generally only necessary that the negligible sets form an ideal; that is, that the empty set be negligible, the union of two negligible sets be negligible, and any subset of a negligible set be negligible. For some purposes, we also need this ideal to be a sigma-ideal, so that countable unions of negligible sets are also negligible. If and are both ideals of subsets of the same set , then one may speak of ''-negligible'' and ''-negligible'' subsets. The opposite of a negligible set is a generic property, which has various forms. Examples Let ''X'' be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
