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:
“This declared indifference, but as I must think, covert real zeal for the spread of slavery, I can not but hate. I hate it because of the monstrous injustice of slavery itself. I hate it because it deprives our republican example of its just influence in the world ... and especially because it forces so many really good men amongst ourselves into an open war with the very fundamental principles of civil liberty.”
—Abraham Lincoln (1809–1865)
“What I call middle-class society is any society that becomes rigidified in predetermined forms, forbidding all evolution, all gains, all progress, all discovery. I call middle-class a closed society in which life has no taste, in which the air is tainted, in which ideas and men are corrupt. And I think that a man who takes a stand against this death is in a sense a revolutionary.”
—Frantz Fanon (1925–1961)
“And at least you know
That maps are of time, not place, so far as the army
Happens to be concerned—the reason being,
Is one which need not delay us.”
—Henry Reed (1914–1986)