Homeomorphism

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Homeomorphism

Topological equivalence redirects here; see also topological equivalence (dynamical systems).

In the mathematical
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
field of topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, a homeomorphism or topological isomorphism (from the Greek
Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
words ?µ???? (homoios) = similar and µ??f? (morphe) = shape = form (Latin deformation of morphe)) is a bicontinuous function
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
between two topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
s.

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Encyclopedia

Topological equivalence redirects here; see also topological equivalence (dynamical systems).

In the mathematical
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
field of topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, a homeomorphism or topological isomorphism (from the Greek
Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
words ?µ???? (homoios) = similar and µ??f? (morphe) = shape = form (Latin deformation of morphe)) is a bicontinuous function
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
between two topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
s. Homeomorphisms are the isomorphism
Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
s in the category of topological spaces
Category of topological spaces

In mathematics, the category of topological spaces, often denoted Top, is the category whose object s are topological spaces and whose morphisms are continuous maps....
— that is, they are the mappings
Map (mathematics)

In mathematics and related technical fields, the term map or mapping is often a synonym for Function . Thus, for example, a partial map is a partial function, and a total map is a total function....
which preserve all the topological properties
Topological property

In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms....
of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.

Roughly speaking, a topological space is a geometric
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
and a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
are homeomorphic to each other, but a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
and a donut
Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
are not. An often-repeated joke is that topologists can't tell the coffee cup from which they are drinking from the donut they are eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

Topology is the study of those properties of objects that do not change when homeomorphisms are applied. As Henri Poincaré
Henri Poincaré

Jules Henri Poincar? was a French mathematician and theoretical physicist, and a philosophy of science. Poincar? is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime....
famously said, mathematics is not the study of objects, but instead, the relations (isomorphisms for instance) between them.

Definition
A function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
f: X → Y between two topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
s (X, T_X) and (Y, T_Y) is called a homeomorphism if it has the following properties:

f is a bijection
Bijection

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y....
(1-1
1-1

1-1 may refer to:* Injective function* Schweizer SGP 1-1, glider...
and onto),
f is continuous,
the inverse function
Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
f⁻¹ is continuous (f is an open mapping).

A function with these three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic. A self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form an equivalence relation
Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
on the class
Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of Set which can be unambiguously defined by a property that all its members share....
of all topological spaces. The resulting equivalence classes
Equivalence class

In mathematics, given a Set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:...
are called homeomorphism classes.

Examples

* The unit 2-disc
Ball (mathematics)

In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general....
D² and the unit square
Unit square

The unit square is a square with all of the side lengths equalling 1....
in R² are homeomorphic.

The open interval
Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
(-1, 1) is homeomorphic to the real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s R.

The product space
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....
S¹
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
× S¹ and the two-dimension
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al torus
Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
are homeomorphic.

Every uniform isomorphism
Uniform isomorphism

In the mathematics field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform property....
and isometric isomorphism is a homeomorphism.

Any 2-sphere with a single point removed is homeomorphic to the set of all points in R² (a 2-dimensional plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
).

Let be a commutative ring with unity and let be a mutiplicative subset of . Then Spec is homeomorphic to

and are not homeomorphic for

An example of a continuous bijection that is not a homeomorphism is the map that takes the half-open interval and wraps it around the circle. In this case the inverse - although it exists - fails to be continuous. The primage of certain sets which are actual open in the relative topology of the half-open interval are not open in the more natural topology of the circle (they are half-open intervals).

Properties

Two homeomorphic spaces share the same topological properties
Topological property

In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms....
. For example, if one of them is compact
Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
, then the other is as well; if one of them is connected
Connectedness

In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected....
, then the other is as well; if one of them is Hausdorff
Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhood ....
, then the other is as well; their homology groups will coincide. Note however that this does not extend to properties defined via a metric
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
; there are metric spaces which are homeomorphic even though one of them is complete and the other is not.

A homeomorphism is simultaneously an open mapping and a closed mapping, that is it maps open set
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
s to open sets and closed set
Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
s to closed sets.

Every self-homeomorphism in can be extended to a self-homeomorphism of the whole disk (Alexander's Trick
Alexander's trick

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

Informal discussion
The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly — it may not be obvious from the description above that deforming a line segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts.

This characterization of a homeomorphism often leads to confusion with the concept of homotopy
Homotopy

In topology, two continuous function Function from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions....
, which is actually defined as a continuous deformation, but from one function to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space X correspond to which points on Y — one just follows them as X deforms. In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: homotopy equivalence.

There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an isotopy
Homotopy

In topology, two continuous function Function from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions....
between the identity map
Identity map

An identity map is a database access design pattern used to improve performance by providing a context-specific in-memory cache to prevent duplicate retrieval of the same object data from the database....
on X and the homeomorphism from X to Y.

See also

Local homeomorphism
Local homeomorphism

In mathematics, more specifically topology, a local homeomorphism is intuitively a function, f, between topological spaces that preserves local structure....
Diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
Uniform isomorphism
Uniform isomorphism

In the mathematics field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform property....
is an isomorphism between uniform spaces
Isometric isomorphism is an isomorphism between metric spaces
Dehn twist
Dehn twist

In geometric topology, a branch of mathematics, a Dehn twist is a certain type of homeomorphism of a surface ....
Homeomorphism (graph theory)
Homeomorphism (graph theory)

In graph theory, two graphs and are homeomorphic if there is an isomorphism from some subdivision of to some subdivision of . If the edges of a graph are thought of as lines drawn from one vertex to another , then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphism in the...
(closely related to graph subdivision)
Isotopy
Mapping class group
Mapping class group

In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space....

External links