Open and Closed Maps - Open and Closed Mapping Theorems

Open and Closed Mapping Theorems

It is useful to have conditions for determining when a map is open or closed. The following are some results along these lines.

The closed map lemma states that every continuous function f : X → Y from a compact space X to a Hausdorff space Y is closed and proper (i.e. preimages of compact sets are compact). A variant of this result states that if a continuous function between locally compact Hausdorff spaces is proper, then it is also closed.

In functional analysis, the open mapping theorem states that every surjective continuous linear operator between Banach spaces is an open map.

In complex analysis, the identically named open mapping theorem states that every non-constant holomorphic function defined on a connected open subset of the complex plane is an open map.

The invariance of domain theorem states that a continuous and locally injective function between two n-dimensional topological manifolds must be open.