Topological equivalence redirects here; see also topological equivalence (dynamical systems). In Mathematics, two functions are said to be topologically conjugate to one another if there exists a Homeomorphism that will conjugate the one into the other

A continuous deformation between a coffee mug and a donut illustrating that they are homeomorphic. A mug is a sturdily built type of cup often used for drinking hot beverages such as Coffee, Tea, or Hot chocolate. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar But there does not need to be a continuous deformation for two spaces to be homeomorphic.

In the mathematical field of topology, a homeomorphism or topological isomorphism (from the Greek words homoios = similar and μορφή (morphē) = shape = form (Latin deformation of morphe)) is a special isomorphism between topological spaces which respects topological properties. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is Two spaces with a homeomorphism between them are called homeomorphic. From a topological viewpoint they are the same.

Roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar 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.

Intuitively, a homeomorphism maps points in the first object that are "close together" to points in the second object that are close together, and points in the first object that are not close together to points in the second object that are not close together. Topology is the study of those properties of objects that do not change when homeomorphisms are applied.

Definition

A function f between two topological spaces X and Y is called a homeomorphism if it has the following properties:

• f is a bijection (1-1 and onto),
• f is continuous,
• the inverse function f ⁻¹ is continuous (f is an open mapping). The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets

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 on the class of all topological spaces. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously The resulting equivalence classes are called homeomorphism classes. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X

Examples

A trefoil knot is homeomorphic to a torus. In Knot theory, the trefoil knot is the simplest nontrivial knot. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar While this may seem illogical, in four dimensions they can easily be deformed continuously.

• The unit 2-disc D² and the unit square in R² are homeomorphic. 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 The unit square is a square with all of the side lengths equalling 1

• The open interval (−1, 1) is homeomorphic to the real numbers R. In Mathematics, an 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 In Mathematics, the real numbers may be described informally in several different ways

• The product space S¹ × S¹ and the two-dimensional torus are homeomorphic. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar

• Every uniform isomorphism and isometric isomorphism is a homeomorphism. In the mathematical field of Topology a uniform isomorphism or uniform homeomorphism is a special Isomorphism between Uniform spaces For the Mechanical engineering and Architecture usage see Isometric projection.

• Any 2-sphere with a single point removed is homeomorphic to the set of all points in R² (a 2-dimensional plane). "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

• and are not homeomorphic for

Notes

The third requirement, that f ⁻¹ be continuous, is essential. Consider for instance the function f : [0, 2π) → S¹ defined by f(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism.

Homeomorphisms are the isomorphisms in the category of topological spaces. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms X → X forms a group, called the homeomorphism group of X, often denoted Homeo(X). In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

For some purposes, the homeomorphism group happens to be too big, but by means of the isotopy relation, one can reduce this group to the mapping class group. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics, in the sub-field of Geometric topology, the mapping class group is an important algebraic invariant of a Topological space.

Properties

• Two homeomorphic spaces share the same topological properties. In Topology and related areas of Mathematics a topological property or topological invariant is a property of a Topological space which is For example, if one of them is compact, then the other is as well; if one of them is connected, then the other is as well; if one of them is Hausdorff, then the other is as well; their homology groups will coincide. In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is Note however that this does not extend to properties defined via a metric; there are metric spaces which are homeomorphic even though one of them is complete and the other is not. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has

• A homeomorphism is simultaneously an open mapping and a closed mapping, that is it maps open sets to open sets and closed sets to closed sets. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Topology and related branches of Mathematics, a closed set is a set whose complement is open.

• Every self-homeomorphism in S¹ can be extended to a self-homeomorphism of the whole disk D² (Alexander's Trick). Alexander's trick, also known as the Alexander trick, is a basic result in Geometric topology, named after J

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 to a point is impermissible, for instance. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points 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, which is actually defined as a continuous deformation, but from one function to another, rather than one space to another. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical 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. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical

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 between the identity map on X and the homeomorphism from X to Y. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical This article is about the Identity Map software design pattern

See also

• Local homeomorphism
• Diffeomorphism
• Uniform isomorphism is an isomorphism between uniform spaces
• Isometric isomorphism is an isomorphism between metric spaces
• Dehn twist
• Homeomorphism (graph theory) (closely related to graph subdivision)
• Isotopy
• Mapping class group

External links

• Homeomorphism on PlanetMath