Homeomorphism

In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré in 1895. In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré in 1895. Very 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. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations are not homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms are not continuous deformations, such as the homeomorphism between a trefoil knot and a circle. An often-repeated mathematical joke is that topologists can't tell the difference between a coffee cup and a donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle. A function f : X → Y {displaystyle f:X o Y} between two topological spaces is a homeomorphism if it has the following properties: A homeomorphism is sometimes called a bicontinuous function. If such a function exists, X {displaystyle X} and Y {displaystyle Y} are homeomorphic. A self-homeomorphism is a homeomorphism from a topological space onto itself. 'Being homeomorphic' is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes. The third requirement, that f − 1 { extstyle f^{-1}} be continuous, is essential. Consider for instance the function f : [ 0 , 2 π ) → S 1 { extstyle f:[0,2pi ) o S^{1}} (the unit circle in R 2 { extstyle mathbb {R} ^{2}} ) defined by f ( ϕ ) = ( cos ⁡ ϕ , sin ⁡ ϕ ) { extstyle f(phi )=(cos phi ,sin phi )} . This function is bijective and continuous, but not a homeomorphism ( S 1 { extstyle S^{1}} is compact but [ 0 , 2 π ) { extstyle [0,2pi )} is not). The function f − 1 { extstyle f^{-1}} is not continuous at the point ( 1 , 0 ) { extstyle (1,0)} , because although f − 1 { extstyle f^{-1}} maps ( 1 , 0 ) { extstyle (1,0)} to 0 { extstyle 0} , any neighbourhood of this point also includes points that the function maps close to 2 π { extstyle 2pi } , but the points it maps to numbers in between lie outside the neighbourhood. Homeomorphisms are the isomorphisms in the category of topological spaces. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms X → X { extstyle X o X} forms a group, called the homeomorphism group of X, often denoted Homeo ( X ) { extstyle { ext{Homeo}}(X)} . This group can be given a topology, such as the compact-open topology, which under certain assumptions makes it a topological group. 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. Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them, Homeo ( X , Y ) , { extstyle { ext{Homeo}}(X,Y),} is a torsor for the homeomorphism groups Homeo ( X ) { extstyle { ext{Homeo}}(X)} and Homeo ( Y ) { extstyle { ext{Homeo}}(Y)} , and, given a specific homeomorphism between X {displaystyle X} and Y {displaystyle Y} , all three sets are identified.