Identification

Identification is the construction of a quotient space by treating specified points of a topological space as the same point. An equivalence relation specifies which points are to be identified. Each equivalence class becomes a single point of the new space, with the quotient topology (Munkres 2000, pp. 137-142). This operation is often described as gluing.

For example, identifying the two endpoints of the closed interval produces a circle. The endpoints become a single point, while every interior point remains distinct. Identifying corresponding points on each pair of opposite sides of a square by translation gives a torus. The analogous identification of opposite faces of a cube gives the three-torus.