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:
“When the inhabitants of some sequestered island first descry the “big canoe” of the European rolling through the blue waters towards their shores, they rush down to the beach in crowds, and with open arms stand ready to embrace the strangers. Fatal embrace! They fold to their bosoms the vipers whose sting is destined to poison all their joys; and the instinctive feeling of love within their breasts is soon converted into the bitterest hate.”
—Herman Melville (1819–1891)
“No domain of nature is quite closed to man at all times.”
—Henry David Thoreau (1817–1862)