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:
“In the greatest confusion there is still an open channel to the soul. It may be difficult to find because by midlife it is overgrown, and some of the wildest thickets that surround it grow out of what we describe as our education. But the channel is always there, and it is our business to keep it open, to have access to the deepest part of ourselves.”
—Saul Bellow (b. 1915)
“Pray but one prayer for me ‘twixt thy closed lips,
Think but one thought of me up in the stars.”
—William Morris (1834–1896)