Open map

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.