Homeomorphism: Difference between revisions
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In [[mathematics]], a '''homeomorphism''' is a [[function]] that maps one [[topological space]] to another with the property that it is [[bijective function|bijective]] and both the function and its [[inverse function|inverse]] are [[continuity#Continuous function|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 [[ | In [[mathematics]], a '''homeomorphism''' is a [[function]] that maps one [[topological space]] to another with the property that it is [[bijective function|bijective]] and both the function and its [[inverse function|inverse]] are [[continuity#Continuous function|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== | ==Formal definition== |
Revision as of 18:26, 12 October 2007
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.