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:
“The open frontier, the hardships of homesteading from scratch, the wealth of natural resources, the whole vast challenge of a continent waiting to be exploited, combined to produce a prevailing materialism and an American drive bent as much, if not more, on money, property, and power than was true of the Old World from which we had fled.”
—Barbara Tuchman (1912–1989)
“Had I made capital on my prettiness, I should have closed the doors of public employment to women for many a year, by the very means which now makes them weak, underpaid competitors in the great workshop of the world.”
—Jane Grey Swisshelm (1815–1884)
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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)