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:
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“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)