Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or 'gluing together' certain points of a given topological space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. The quotient topology consists of all sets with an open preimage under the canonical projection map that maps each element to its equivalence class. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or 'gluing together' certain points of a given topological space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. The quotient topology consists of all sets with an open preimage under the canonical projection map that maps each element to its equivalence class. Let (X, τX) be a topological space, and let ~ be an equivalence relation on X. The quotient space, Y = X / ~ 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 surjective map q : X → X / ~, 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 map q. A map f : X → Y {displaystyle f:X o Y} is a quotient map (sometimes called an identification map) if it is surjective, and a subset U of Y is open if and only if f − 1 ( U ) {displaystyle f^{-1}(U)} is open. Equivalently, f {displaystyle f} is a quotient map if it is onto and Y {displaystyle Y} is equipped with the final topology with respect to f {displaystyle f} . Given an equivalence relation ∼ {displaystyle sim } on X {displaystyle X} , the canonical map q : X → X / ∼ {displaystyle q:X o X/{sim }} is a quotient map. Note: The notation R/Z is somewhat ambiguous. If Z is understood to be a group acting on R via addition, then the quotient is the circle. However, if Z is thought of as a subspace of R, then the quotient is a countably infinite bouquet of circles joined at a single point. Quotient maps q : X → Y are characterized among surjective maps by the following property: if Z is any topological space and f : Y → Z is any function, then f is continuous if and only if f ∘ q is continuous.