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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. | ||
Revision as of 16:30, 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.