Open And Closed Maps
In topology, an open map is a function between two topological spaces which maps open sets to open sets. That is, a function f : X → Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a closed map is a function which maps closed sets to closed sets. (The concept of a closed map should not be confused with that of a closed operator.)
Neither open nor closed maps are required to be continuous. Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that, by definition, a function f : X → Y is continuous if the preimage of every open set of Y is open in X. (Equivalently, if the preimage of every closed set of Y is closed in X).
Read more about Open And Closed Maps: Examples, Properties, Open and Closed Mapping Theorems
Famous quotes containing the words open, closed and/or maps:
“Outside the open window
The morning air is all awash with angels.
Some are in bed-sheets, some are in blouses,
Some are in smocks: but truly there they are.”
—Richard Wilbur (b. 1921)
“Thus piteously Love closed what he begat:
The union of this ever-diverse pair!
These two were rapid falcons in a snare,
Condemned to do the flitting of the bat.”
—George Meredith (1828–1909)
“And now good morrow to our waking souls,
Which watch not one another out of fear;
For love all love of other sights controls,
And makes one little room an everywhere.
Let sea-discoverers to new worlds have gone,
Let maps to other, worlds on worlds have shown,
Let us possess one world; each hath one, and is one.”
—John Donne (1572–1631)