  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. == definition == let  be a topological space, and let ~ be an equivalence relation on x. the quotient space, y = x/\!\!\sim is defined to be the set of equivalence classes of elements of : :y = \{ [x] : x \in x \} = \{\{v \in x : v \sim x\} : x \in 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: :\tau_y = \left\{ u \subseteq y : \bigcup u = \left( \bigcup_{ [a] \in u} [a] \right) \in \tau_x \right\}. equivalently, we can define them to be those sets with an open preimage under the surjective map q: x \to x/\!\!\sim which sends a point in x to the equivalence class containing it: :\tau_y=\left\{ u \subseteq y : q^{-1}(u) \in \tau_x \right\}. the quotient topology is the final topology on the quotient space with respect to the map q. == quotient map == a map f: x\to y is a quotient map if it is surjective, continuous, and a subset u of y is open if and only if f^{-1}(u) is open. equivalently, f is a quotient map if it is onto and y is equipped with the final topology with respect to f. given an equivalence relation \sim on x, the canonical map q: x \to x/\!\!\sim is a quotient map. == examples == *gluing. often, topologists talk of gluing points together. if x is a topological space and points x,y \in x are to be "glued", then what is meant is that we are to consider the quotient space obtained from the equivalence relation a ~ b if and only if a = b or a = x, b = y (or a = y, b = x). *consider the unit square i2 = [0,1]&times;[0,1] and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. then i2/~ is homeomorphic to the unit sphere s2. *adjunction space. more generally, suppose x is a space and a is a subspace of x. one can i
General_topology Group_actions Topology