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, 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)
“A temple, you know, was anciently “an open place without a roof,” whose walls served merely to shut out the world and direct the mind toward heaven; but a modern meeting-house shuts out the heavens, while it crowds the world into still closer quarters.”
—Henry David Thoreau (1817–1862)
“Night hath closed all in her cloak,
Twinkling stars love-thoughts provoke,
Danger hence good care doth keep,
Jealousy itself doth sleep;”
—Sir Philip Sidney (1554–1586)