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Semiregular Space
A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a base for the topology. Examples and sufficient conditions Every regular space is semiregular, and every topological space may be embedded into a semiregular space.. The space X = \Reals^2 \cup \ with the double origin topology In mathematics, more specifically general topology, the double origin topology is an example of a topology given to the plane R2 with an extra point, say 0*, added. In this case, the double origin topology gives a topology on the set , where ∐ d ... and the Arens squareSteen & Seebach, example #80 are examples of spaces that are Hausdorff semiregular, but not regular. See also * Notes References * Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition). * Properties of topological spaces Separation ... [...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] |
Regular Open Set
A subset S of a topological space X is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if \operatorname(\overline) = S or, equivalently, if \partial(\overline)=\partial S, where \operatorname S, \overline and \partial S denote, respectively, the interior, closure and boundary of S.Steen & Seebach, p. 6 A subset S of X is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if \overline = S or, equivalently, if \partial(\operatornameS)=\partial S. Examples If \Reals has its usual Euclidean topology then the open set S = (0,1) \cup (1,2) is not a regular open set, since \operatorname(\overline) = (0,2) \neq S. Every open interval in \R is a regular open set and every non-degenerate closed interval (that is, a closed interval containing at least two distinct points) is a regular closed set. A singleton \ is a closed subset of \R but not a regular closed set because its interior is the empt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (topology)
In mathematics, a base (or basis) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open intervals in the real number line \R is a basis for the Euclidean topology on \R because every open interval is an open set, and also every open subset of \R can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X form a base for a topology on X. Under some cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Space
In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means "a regular Hausdorff space". These conditions are examples of separation axioms. Definitions A topological space ''X'' is a regular space if, given any closed set ''F'' and any point ''x'' that does not belong to ''F'', there exists a neighbourhood ''U'' of ''x'' and a neighbourhood ''V'' of ''F'' that are disjoint. Concisely put, it must be possible to separate ''x'' and ''F'' with disjoint neighborhoods. A or is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T3 if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Origin Topology
In mathematics, more specifically general topology, the double origin topology is an example of a topology given to the plane R2 with an extra point, say 0*, added. In this case, the double origin topology gives a topology on the set , where ∐ denotes the disjoint union. Construction Given a point ''x'' belonging to ''X'', such that and , the neighbourhoods of ''x'' are those given by the standard metric topology on We define a countably infinite basis of neighbourhoods about the point 0 and about the additional point 0*. For the point 0, the basis, indexed by ''n'', is defined to be: : \ N(0,n) = \ \cup \ . In a similar way, the basis of neighbourhoods of 0* is defined to be: :N(0^*,n) = \ \cup \ . Properties The space , along with the double origin topology is an example of a Hausdorff space, although it is not completely Hausdorff. In terms of compactness, the space , along with the double origin topology fails to be either compact, paracompact or locally compact, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arens Square
In mathematics, the Arens square is a topological space, named for Richard Friederich Arens. Its role is mainly to serve as a counterexample. Definition The Arens square is the topological space (X,\tau), where :X=((0,1)^2\cap\mathbb^2)\cup\\cup\\cup\ The topology \tau is defined from the following basis. Every point of (0,1)^2\cap\mathbb^2 is given the local basis of relatively open sets inherited from the Euclidean topology on (0,1)^2. The remaining points of X are given the local bases *U_n(0,0)=\\cup\ *U_n(1,0)=\\cup\ *U_n(1/2,r\sqrt)=\ Properties The space (X,\tau) is: # T2½, since neither points of (0,1)^2\cap\mathbb^2, nor (0,0), nor (0,1) can have the same second coordinate as a point of the form (1/2,r\sqrt), for r\in\mathbb. # not T3 or T3½, since for (0,0)\in U_n(0,0) there is no open set U such that (0,0)\in U\subset \overline\subset U_n(0,0) since \overline must include a point whose first coordinate is 1/4, but no such point exists in U_n(0,0) for any n\in\ma ... [...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] |
Properties Of Topological Spaces
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an object *Material properties, properties by which the benefits of one material versus another can be assessed *Chemical property, a material's properties that becomes evident during a chemical reaction * Physical property, any property that is measurable whose value describes a state of a physical system *Semantic property *Thermodynamic properties, in thermodynamics and materials science, intensive and extensive physical properties of substances *Mental property, a property of the mind studied by many sciences and parasciences Computer science * Property (programming), a type of class member in object-oriented programming * .properties, a Java Properties File to store program settings as name-value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
