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:
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“One definition of man is an intelligence served by organs.”
—Ralph Waldo Emerson (18031882)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (18391894)