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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. Widely known topologies * The Baire space − \N^ with the product topology, where \N denotes the natural numbers endowed with the discrete topology. It is the space of all sequences of natural numbers. * Cantor set − A subset of the closed interval , 1/math> with remarkable properties. ** Cantor dust * Discrete topology − All subsets are open. * Euclidean topology − The natural topology on Euclidean space \Reals^n induced by the Euclidean metric, which is itself induced by the Euclidean norm. ** Real line − \Reals ** Space-filling curve ** Unit interval − , 1/math> * Extended real number line * Hilbert cube − , 1/1\times , 1/2\times , 1/3\times \cdots with t ... [...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 space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space-filling Curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called ''Peano curves'', but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano. Definition Intuitively, a curve in two or three (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a ''curve'': In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a ''pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-torus
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, \mathbb^3 = S^1 \times S^1 \times S^1. In contrast, the usual torus is the Cartesian product of only two circles. The 3-torus is a three-dimensional compact manifold with no boundary. It can be obtained by "gluing" the three pairs of opposite faces of a cube, where being "glued" can be intuitively understood to mean that when a particle moving in the interior of the cube reaches a point on a face, it goes through it and appears to come forth from the corresponding point on the opposite face, producing periodic boundary conditions. Gluing only one pair of opposite faces produces a solid torus while gluing two of these pairs produces the solid space between two nested tori. In 1984, Alexei Starobinsky and Yakov Borisovich Zel'dovich Yakov Borisovich Zeldovich ( be, Я́каў Бары́савіч Зяльдо́віч, russian: Я́ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a '' solid torus'', which is formed by r ... [...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 = \ 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 particular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Two-point Set
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 = \ 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 particu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sierpiński Space
In mathematics, the Sierpiński space (or the connected two-point set) 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Projective Line
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line (geometry), line that has been historically introduced to solve a problem set by visual perspective (visual), perspective: two parallel lines do not intersect but seem to intersect "at infinity". For solving this problem, points at infinity have been introduced, in such a way that in a real projective plane, two distinct projective lines meet in exactly one point. The set of these points at infinity, the "horizon" of the visual perspective in the plane, is a real projective line. It is the set of directions emanating from an observer situated at any point, with opposite directions identified. An example of a real projective line is the projectively extended real line, which is often called ''the'' projective line. Formally, a real projective line P(R) is defined as the set of all one-dimensional linear subspaces of a two-dimensional vector space ove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Bottle
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. While a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary. The concept of a Klein bottle was first described in 1882 by the German mathematician Felix Klein. Construction The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding red and blue edges with the arrows matching, as in the diagrams below. Note that this is an "abstract" gluing in the sense that trying to realize ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero. Details The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space ''X'' with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space. Other properties of an indiscrete space ''X''—many of which are quite unusual—include: * The only closed sets are the empty set and ''X''. * The only possible basis of ''X'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indiscrete Topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero. Details The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space ''X'' with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space. Other properties of an indiscrete space ''X''—many of which are quite unusual—include: * The only closed sets are the empty set and ''X''. * The only possible basis of ''X'' is . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product. Definition Throughout, I will be some non-empty index set and for every index i \in I, let X_i be a topological space. Denote the Cartesian product of the sets X_i by X := \prod X_ := \prod_ X_i and for every index i \in I, denote the i-th by \begin p_i :\;&& \prod_ X_j &&\;\to\; & X_i \\ .3ex && \left(x_j\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
