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:
“It is open to a war resister to judge between the combatants and wish success to the one who has justice on his side. By so judging he is more likely to bring peace between the two than by remaining a mere spectator.”
—Mohandas K. Gandhi (1869–1948)
“For a long time, I went to bed early. Sometimes, my candle barely put out, my eyes closed so quickly that I did not have the time to say to myself: “I am falling asleep”.”
—Marcel Proust (1871–1922)
“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)