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List Of Topologies
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property. Discrete and indiscrete * Discrete topology − All subsets are open. * Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open. Cardinality and ordinals * Cocountable topology ** Given a topological space (X, \tau), the '' '' on X is the topology having as a subbasis the union of and the family of all subsets of X whose complements in X are countable. * Cofinite topology * Double-pointed cofinite topology * Ordinal number topology * Pseudo-arc * Ran space * Tychonoff plank Finite spaces * Discrete two-point space − The simplest example of a totally disconnected discrete space. * Finite topological space * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology (structure)
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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the ''only'' connected subsets. An important example of a totally disconnected space is the Cantor set, which is homeomorphic to the set of ''p''-adic integers. Another example, playing a key role in algebraic number theory, is the field of ''p''-adic numbers. Definition A topological space X is totally disconnected if the connected components in X are the one-point sets. Analogously, a topological space X is totally path-disconnected if all path-components in X are the one-point sets. Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents are singletons. That is, a topological space X is totally separated if for every x\i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Progression Topologies
In general topology and number theory, branches of mathematics, one can define various topologies on the set \mathbb of integers or the set \mathbb_ of positive integers by taking as a base a suitable collection of arithmetic progressions, sequences of the form \ or \. The open sets will then be unions of arithmetic progressions in the collection. Three examples are the Furstenberg topology on \mathbb, and the Golomb topology and the Kirch topology on \mathbb_. Precise definitions are given below. Hillel Furstenberg introduced the first topology in order to provide a "topological" proof of the infinitude of the set of primes. The second topology was studied by Solomon Golomb and provides an example of a countably infinite Hausdorff space that is connected. The third topology, introduced by A.M. Kirch, is an example of a countably infinite Hausdorff space that is both connected and locally connected. These topologies also have interesting separation and homogeneity propertie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arens–Fort Space
In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr. Definition The Arens–Fort space is the topological space (X,\tau) where X is the set of ordered pairs of non-negative integers (m, n). A subset U \subseteq X is open, that is, belongs to \tau, if and only if: * U does not contain (0, 0), or * U contains (0, 0) and also all but a finite number of points of all but a finite number of columns, where a column is a set \ with 0 \leq m \in \mathbb fixed. In other words, an open set is only "allowed" to contain (0, 0) if only a finite number of its columns contain significant gaps, where a gap in a column is significant if it omits an infinite number of points. Properties It is * Hausdorff * regular * normal It is not: * second-countable * first-countable * metrizable * compact * sequential * Fréchet–Urysohn There is no sequence in X \setminus \ that converges to (0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particular Point Topology
In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collection :T = \ \cup \ of subsets of ''X'' is the particular point topology on ''X''. There are a variety of cases that are individually named: * If ''X'' has two points, the particular point topology on ''X'' is the Sierpiński space. * If ''X'' is finite (with at least 3 points), the topology on ''X'' is called the finite particular point topology. * If ''X'' is countably infinite, the topology on ''X'' is called the countable particular point topology. * If ''X'' is uncountable, the topology on ''X'' is called the uncountable particular point topology. A generalization of the particular point topology is the closed extension topology. In the case when ''X'' \ has the discrete topology, the closed extension topology is the same as the pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sierpiński Space
In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński. The Sierpiński space has important relations to the theory of computation and semantics, because it is the classifying space for open sets in the Scott topology. Definition and fundamental properties Explicitly, the Sierpiński space is a topological space ''S'' whose underlying point set is \ and whose open sets are \. The closed sets are \. So the singleton set \ is closed and the set \ is open (\varnothing = \ is the empty set). The closure operator on ''S'' is determined by \overline = \, \qquad \overline = \. A finite topological space is also uniquely determined by its specialization preorder. For the Sierpiński space this preorder is actually a partial order and given by 0 \leq 0, \qquad 0 \leq 1, \qquad 1 \leq 1. Topol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a Disk (mathematics), disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus (mathematics), Annulus: a ring-shaped object, the region bounded by two concentric circles. * Circular arc, Arc: any Connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov Space
In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing the other. In a T0 space, all points are topologically distinguishable. This condition, called the T0 condition, is the weakest of the separation axioms. Nearly all topological spaces normally studied in mathematics are T0 spaces. In particular, all T1 spaces, i.e., all spaces in which for every pair of distinct points, each has a neighborhood not containing the other, are T0 spaces. This includes all T2 (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction, every sober space (which may not be T1) is T0; this includes the underlying topological space of any scheme. Given any topological space one can construct a T0 space by identifying topologically indistinguisha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called ''Tychonoff separation axioms'', after Andrey Tychonoff. The separation axioms are not fundamental axioms like those of Zermelo–Fraenkel set theory, set theory, but rather defining properties which may be specified to distinguish certain types of topological spaces. The separation axioms are denoted with the letter "T" after the German language, German ''Trennungsaxiom'' ("separation axiom"), and increasing numerical subscripts denote stronger and stronger properties. The precise definitions of the history of the separation axioms, separation axioms have varied over time. Especially in older literature, different authors might have different definitions of each condition. Preliminary definitions Before we define the separation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Topological Space
In mathematics, a finite topological space is a topological space for which the underlying set (mathematics), point set is finite set, finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions". Topologies on a finite set Let X be a finite set. A topology (structure), topology on X is a subset \tau of P(X) (the power set of X ) such that # \varnothing \in \tau and X\in \tau . # if U, V \in \tau then U \cup V \in \tau . # if U, V \in \tau then U \cap V \in \tau . In other words, a subset \tau of P(X) is a topology if \tau contains both \varnothing and X and is closed under arbitrary union (set theory), unions and intersection (set theory), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
