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In [[general topology]], an '''open map''' is a [[function (mathematics)|function]] on a [[topological space]] which maps every [[open set]] in the domain to an open set in the image. | In [[general topology]], an '''open map''' is a [[function (mathematics)|function]] on a [[topological space]] which maps every [[open set]] in the domain to an open set in the image. | ||
A [[homeomorphism]] may be defined as a [[continuous map|continuous]] open [[bijection]]. | A [[homeomorphism]] may be defined as a [[continuous map|continuous]] open [[bijection]]. | ||
==Open mapping theorem== | |||
The '''open mapping theorem''' states that under suitable conditions a differentiable function may be an open map. | |||
''Open mapping theorem for real functions''. Let ''f'' be a function from an open domain ''D'' in '''R'''<sup>''n''</sup> to '''R'''<sup>''n''</sup> which is differentiable and has [[non-singular map|non-singular]] [[Derivative#Multivariable calculus|derivative]] [[non-singular map|non-singular]] in ''D''. Then ''f'' is an open map on ''D''. | |||
''Open mapping theorem for complex functions''. Let ''f'' be a non-constant [[holomorphic function]] on an open domain ''D'' in the [[complex plane]]. Then ''f'' is an open map on ''D''. | |||
==References== | |||
* {{cite book | author=Tom M. Apostol | title=Mathematical Analysis | edition=2nd ed | publisher=Addison-Wesley | year=1974 | pages=371,454 }} | |||
* {{cite book | author=J.L. Kelley | authorlink=John L. Kelley | title=General topology | publisher=van Nostrand | year= 1955 | pages=90 }} | |||
Latest revision as of 17:30, 7 February 2009
In general topology, an open map is a function on a topological space which maps every open set in the domain to an open set in the image.
A homeomorphism may be defined as a continuous open bijection.
Open mapping theorem
The open mapping theorem states that under suitable conditions a differentiable function may be an open map.
Open mapping theorem for real functions. Let f be a function from an open domain D in Rn to Rn which is differentiable and has non-singular derivative non-singular in D. Then f is an open map on D.
Open mapping theorem for complex functions. Let f be a non-constant holomorphic function on an open domain D in the complex plane. Then f is an open map on D.
References
- Tom M. Apostol (1974). Mathematical Analysis, 2nd ed. Addison-Wesley, 371,454.
- J.L. Kelley (1955). General topology. van Nostrand, 90.