Quotient topology

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Quotient topology is a concept in the branch of mathematics known as topology.


Let be a topological space, and a set, and let be a surjection. The quotient topology on induced by is the topology whose open sets are the sets such that is an open set in .[1]


Quotient topologies can often be visualized as gluing elements of a topological space together.

Let with the usual topology (as a subspace of the reals), , and be given by and for . Then 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.


  1. C. Adams and R. Franzosa. Introduction to Topology: Pure and Applied. Upper Saddle River, NJ: Pearson Prentice Hall, 2008. p. 89