Homeomorphism

A classic example of homeomorphism: a circle and a bagel ( torus ) are topologically equivalent

Homeomorphism ( Greek ὅμοιος - similar, μορφή - form) is a one-to-one and one-to-one mapping of topological spaces . In other words, this is a bijection connecting the topological structures of two spaces, because, with continuity of the bijection, the images and inverse images of open subsets are open sets that determine the topologies of the corresponding spaces.

The spaces connected by homeomorphism are topologically indistinguishable. We can say that topology, in its general form, studies the properties of objects that are unchanged under homeomorphism.

Only continuous mappings are considered in the category of topological spaces ; therefore, in this category the isomorphism is also a homeomorphism.

Definition

Let be $(X,{\mathcal {T}}_{X})$ ${\ displaystyle (X, {\ mathcal {T}} _ {X})}$ $(X,\mathcal{T}_X)$ and $(Y,{\mathcal {T}}_{Y})$ ${\ displaystyle (Y, {\ mathcal {T}} _ {Y})}$ $(Y,\mathcal{T}_Y)$ - two topological spaces . Function $f:X\to Y$ ${\ displaystyle f: X \ to Y}$ $f:X \to Y$ called a homeomorphism if it is one-to-one , and also $f$ ${\ displaystyle f}$ $f$ and inverse function $f^{-1}$ ${\ displaystyle f ^ {- 1}}$ $f^{-1}$ continuous .

Related Definitions

Spaces $X$ {\ displaystyle X} and $Y$ {\ displaystyle Y} in this case are called homeomorphic or topologically equivalent .
- This relationship is usually denoted by $X\simeq Y$ ${\ displaystyle X \ simeq Y}$ $X\simeq Y$ .

Homeomorphism Theorem

Let be $|a,b|\subset \mathbb {R}$ ${\ displaystyle | a, b | \ subset \ mathbb {R}}$ $|a,b|\subset \mathbb{R}$ - the interval on the number line (open, half-open or closed). Let be $f:|a,b|\to f{\bigl (}|a,b|{\bigr )}\subset \mathbb {R}$ ${\ displaystyle f: | a, b | \ to f {\ bigl (} | a, b | {\ bigr)} \ subset \ mathbb {R}}$ $f:|a,b| \to f\bigl( |a,b| \bigr)\subset \R$ - bijection. Then $f$ ${\ displaystyle f}$ $f$ is a homeomorphism if and only if $f$ ${\ displaystyle f}$ $f$ strictly monotonous and continuous on $|a,b|.$ ${\ displaystyle | a, b |.}$ $|a,b|.$

Example

Arbitrary open interval $(a,b)\subset \mathbb {R}$ ${\ displaystyle (a, b) \ subset \ mathbb {R}}$ $(a,b) \subset \mathbb{R}$ homeomorphic to the whole number line $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ $\mathbb {R}$ . Homeomorphism $f:(a,b)\to \mathbb {R}$ ${\ displaystyle f: (a, b) \ to \ mathbb {R}}$ $f:(a,b) \to \mathbb{R}$ is given, for example, by the formula

f(x)=\mathrm {ctg} \left(\pi {\frac {x-a}{b-a}}\right).

{\ displaystyle f (x) = \ mathrm {ctg} \ left (\ pi {\ frac {xa} {ba}} \ right).}

Interval $(0,\;1)$ ${\ displaystyle (0, \; 1)}$ homeomorphic to the segment $[0,\;1]$ ${\ displaystyle [0, \; 1]}$ in a discrete topology , but not homeomorphic in the standard for a numerical direct topology.

Notes

Literature

Zorich V.A. Mathematical analysis. - M .: Nauka , 1984 .-- T. 2 .-- S. 41.
Vasiliev V.A. Introduction to topology. - M .: FAZIS, 1997. - Issue. 3 .-- xii + 132 s. - (Student Mathematics Library). - ISBN 5-7036-0036-7 .
Timofeeva N.V. Differential geometry and elements of topology . - YAGPU , 2007.
Boltyanskiy V.G. , Efremovich V.A. Clear topology. - M .: Science, 1982. - 160 p.

Links

Gorbov A.I. Homeomorphism // Brockhaus and Efron Encyclopedic Dictionary : 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.