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.
Read more about this topic: Open And Closed Maps
Famous quotes containing the words open and/or closed:
“Meanwhile Snow White held court,
rolling her china-blue doll eyes open and shut
and sometimes referring to her mirror
as women do.”
—Anne Sexton (1928–1974)
“With two sons born eighteen months apart, I operated mainly on automatic pilot through the ceaseless activity of their early childhood. I remember opening the refrigerator late one night and finding a roll of aluminum foil next to a pair of small red tennies. Certain that I was responsible for the refrigerated shoes, I quickly closed the door and ran upstairs to make sure I had put the babies in their cribs instead of the linen closet.”
—Mary Kay Blakely (20th century)