Definition
Let (X,τX) be a topological space, and let ~ be an equivalence relation on X. The quotient space, is defined to be the set of equivalence classes of elements of X:
equipped with the topology where the open sets are defined to be those sets of equivalence classes whose unions are open sets in X:
Equivalently, we can define them to be those sets with an open preimage under the quotient map which sends a point in X to the equivalence class containing it.
The quotient topology is the final topology on the quotient space with respect to the quotient map.
Read more about this topic: Quotient Space
Famous quotes containing the word definition:
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)