Homeomorphism, the Glossary

Table of Contents

1. 62 relations: Alexander's trick, Atlas (topology), Ball (mathematics), Bijection, Category of topological spaces, Circle, Closed set, Compact space, Compact-open topology, Complete metric space, Connectedness, Continuous function, Curve, Differentiable function, Domain of a function, Doughnut, Equivalence class, Equivalence relation, Euclidean space, Function (mathematics), Geometry, Graph of a function, Group (mathematics), Hausdorff space, Henri Poincaré, Homeomorphism group, Homology (mathematics), Homotopy, Identity function, Injective function, Interval (mathematics), Inverse function, Isomorphism, Line segment, Manifold, Map (mathematics), Mathematics, Metric space, Mug, Neighbourhood (mathematics), Neoclassical compound, Number line, Open and closed maps, Open set, Parametric equation, Plane (mathematics), Polar coordinate system, Principal homogeneous space, Real number, Sphere, ... Expand index (12 more) »

2. Homeomorphisms

Alexander's trick

Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander. Homeomorphism and Alexander's trick are homeomorphisms.

See Homeomorphism and Alexander's trick

Atlas (topology)

In mathematics, particularly topology, an atlas is a concept used to describe a manifold.

See Homeomorphism and Atlas (topology)

Ball (mathematics)

In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere.

See Homeomorphism and Ball (mathematics)

Bijection

A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the first set (the domain) is mapped to exactly one element of the second set (the codomain). Homeomorphism and bijection are functions and mappings.

See Homeomorphism and Bijection

Category of topological spaces

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps.

See Homeomorphism and Category of topological spaces

Circle

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre.

See Homeomorphism and Circle

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

See Homeomorphism and Closed set

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.

See Homeomorphism and Compact space

Compact-open topology

In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces.

See Homeomorphism and Compact-open topology

Complete metric space

In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in.

See Homeomorphism and Complete metric space

Connectedness

In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece".

See Homeomorphism and Connectedness

Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. Homeomorphism and continuous function are theory of continuous functions.

See Homeomorphism and Continuous function

Curve

In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.

See Homeomorphism and Curve

Differentiable function

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

See Homeomorphism and Differentiable function

Domain of a function

In mathematics, the domain of a function is the set of inputs accepted by the function. Homeomorphism and domain of a function are functions and mappings.

See Homeomorphism and Domain of a function

Doughnut

A doughnut or donut is a type of pastry made from leavened fried dough.

See Homeomorphism and Doughnut

Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes.

See Homeomorphism and Equivalence class

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

See Homeomorphism and Equivalence relation

Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

See Homeomorphism and Euclidean space

Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of. Homeomorphism and function (mathematics) are functions and mappings.

See Homeomorphism and Function (mathematics)

Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

See Homeomorphism and Geometry

Graph of a function

In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x). Homeomorphism and graph of a function are functions and mappings.

See Homeomorphism and Graph of a function

Group (mathematics)

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

See Homeomorphism and Group (mathematics)

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 that are disjoint from each other.

See Homeomorphism and Hausdorff space

Henri Poincaré

Jules Henri Poincaré (29 April 185417 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

See Homeomorphism and Henri Poincaré

Homeomorphism group

In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation.

See Homeomorphism and Homeomorphism group

Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces.

See Homeomorphism and Homology (mathematics)

Homotopy

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. Homeomorphism and homotopy are theory of continuous functions.

See Homeomorphism and Homotopy

Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. Homeomorphism and identity function are functions and mappings.

See Homeomorphism and Identity function

Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies. Homeomorphism and injective function are functions and mappings.

See Homeomorphism and Injective function

Interval (mathematics)

In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps".

See Homeomorphism and Interval (mathematics)

Inverse function

In mathematics, the inverse function of a function (also called the inverse of) is a function that undoes the operation of.

See Homeomorphism and Inverse function

Isomorphism

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.

See Homeomorphism and Isomorphism

Line segment

In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints.

See Homeomorphism and Line segment

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

See Homeomorphism and Manifold

Map (mathematics)

In mathematics, a map or mapping is a function in its general sense. Homeomorphism and map (mathematics) are functions and mappings.

See Homeomorphism and Map (mathematics)

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Homeomorphism and Mathematics

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.

See Homeomorphism and Metric space

Mug

A mug is a type of cup typically used for drinking hot drinks such as; coffee, hot chocolate, or tea.

See Homeomorphism and Mug

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

See Homeomorphism and Neighbourhood (mathematics)

Neoclassical compound

Neoclassical compounds are compound words composed from combining forms (which act as affixes or stems) derived from classical languages (classical Latin or ancient Greek) roots.

See Homeomorphism and Neoclassical compound

Number line

In elementary mathematics, a number line is a picture of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely.

See Homeomorphism and Number line

Open and closed maps

In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. Homeomorphism and open and closed maps are theory of continuous functions.

See Homeomorphism and Open and closed maps

Open set

In mathematics, an open set is a generalization of an open interval in the real line.

See Homeomorphism and Open set

Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.

See Homeomorphism and Parametric equation

Plane (mathematics)

In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely.

See Homeomorphism and Plane (mathematics)

Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

See Homeomorphism and Polar coordinate system

Principal homogeneous space

In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial.

See Homeomorphism and Principal homogeneous space

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

See Homeomorphism and Real number

Sphere

A sphere (from Greek) is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.

See Homeomorphism and Sphere

Square

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four sides of equal length and four equal angles (90-degree angles, π/2 radian angles, or right angles).

See Homeomorphism and Square

Stereographic projection

In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the projection plane) perpendicular to the diameter through the point.

See Homeomorphism and Stereographic projection

Surjective function

In mathematics, a surjective function (also known as surjection, or onto function) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that. Homeomorphism and surjective function are functions and mappings.

See Homeomorphism and Surjective function

Topological group

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.

See Homeomorphism and Topological group

Topological property

In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Homeomorphism and topological property are homeomorphisms.

See Homeomorphism and Topological property

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.

See Homeomorphism and Topological space

Topology

Topology (from the Greek words, and) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

See Homeomorphism and Topology

Torus

In geometry, a torus (tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle.

See Homeomorphism and Torus

Trefoil knot

In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot.

See Homeomorphism and Trefoil knot

Uniform space

In the mathematical field of topology, a uniform space is a set with additional structure that is used to define uniform properties, such as completeness, uniform continuity and uniform convergence.

See Homeomorphism and Uniform space

Unit circle

In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1.

See Homeomorphism and Unit circle

Unit square

In mathematics, a unit square is a square whose sides have length.

See Homeomorphism and Unit square

See also

Homeomorphisms

• Alexander's trick
• Borel conjecture
• Carathéodory's theorem (conformal mapping)
• Dehn twist
• Diffeomorphisms
• Discontinuous group
• Homeomorphism
• Homeomorphism (graph theory)
• Homeotopy
• Invariance of domain
• Lantern relation
• Local diffeomorphism
• Local homeomorphism
• Local property
• Mapping class group
• Mapping torus
• Nielsen realization problem
• Nielsen–Thurston classification
• Pseudo-Anosov map
• Quasiconformal mapping
• Quasisymmetric map
• Schoenflies problem
• Self-similarity
• Topological conjugacy
• Topological property
• Toronto space
• Uniform isomorphism
• Universal homeomorphism
• Y-homeomorphism

References

[1] https://en.wikipedia.org/wiki/Homeomorphism

Also known as Bi-continuous, Bicontinuous, Bicontinuous function, Bicontinuous function space, Bicontinuous topological space, Homeomorphic, Homeomorphism class, Homeomorphisms, Homoeomorphic, Homoeomorphism, Topological equivalence, Topological isomorphism.

, Square, Stereographic projection, Surjective function, Topological group, Topological property, Topological space, Topology, Torus, Trefoil knot, Uniform space, Unit circle, Unit square.