Open map

In general topology, an open map is a function from a topological space (the domain) to (the same or another) topological space (the codomain) that maps every open set in the domain to an open set in the codomain.

A homeomorphism may be defined as a continuous open bijection.

[edit] 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 continuously differentiable and has non-singular derivative (in other words, non-zero Jacobian) 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.

Some content on this page may previously have appeared on Citizendium.