# 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 **R**^{n} to **R**^{n} 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*.

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