Homeomorphism: Difference between revisions
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==Formal definition== | ==Formal definition== | ||
Let <math>\scriptstyle (X,O_X)</math> and <math>\scriptstyle (Y,O_Y)</math> be topological spaces. A function <math>\scriptstyle f:(X,O_X)\rightarrow (Y,O_Y)</math> is a homeomorphism | Let <math>\scriptstyle (X,O_X)</math> and <math>\scriptstyle (Y,O_Y)</math> be topological spaces. A function <math>\scriptstyle f:(X,O_X)\rightarrow (Y,O_Y)</math> is a homeomorphism between <math>\scriptstyle (X,O_X)</math> and <math>\scriptstyle (Y,O_Y)</math> if it has the following properties: | ||
#f is a bijective function (i.e., it is [[injective function|one-to-one]] and [[surjective function|onto]]) | #f is a bijective function (i.e., it is [[injective function|one-to-one]] and [[surjective function|onto]]) | ||
#f is continuous | #f is continuous |
Revision as of 12:54, 2 November 2008
In mathematics, a homeomorphism is a function that maps one topological space to another with the property that it is bijective and both the function and its inverse are continuous with respect to the associated topologies. A homeomorphism indicates that the two topological spaces are "geometrically" alike, in the sense that points that are "close" in one space are mapped to points which are also "close" in the other, while points that are "distant" are also mapped to points which are also "distant". In differential geometry, this means that one topological space can be deformed into the other by "bending" and "stretching".
Formal definition
Let and be topological spaces. A function is a homeomorphism between and if it has the following properties:
- f is a bijective function (i.e., it is one-to-one and onto)
- f is continuous
- The inverse function is a continuous function.
If some homeomorphism exists between two topological spaces and then they are said to be homeomorphic to one another.