Quotient topologies can often be visualized as gluing elements of a topological space together.
Let X = [0,1] with the usual topology (as a subspace of the reals), A = [0,1), and be given by p(1) = 1 and p(x) = x for . Then A under the quotient topology is homeomorphic to the circle. Indeed, we can visualize what happened as a gluing operation: the two endpoints of the interval were glued together to create a closed loop.
- ↑ C. Adams and R. Franzosa. Introduction to Topology: Pure and Applied. Upper Saddle River, NJ: Pearson Prentice Hall, 2008. p. 89