Quotient topology

In general topology, the quotient topology, or identification topology is defined on the image of a topological space under a function.

Let $(X,{\mathcal {T}})$ be a topological space, and q a surjective function from X onto a set Y. The quotient topology on Y has as open sets those subsets $U$ of $Y$ such that the pre-image $q^{-1}(U)=\{x\in X\mid q(x)\in U\}\in {\mathcal {T}}_{X}$.

The quotient topology has the universal property that it is the finest topology such that q is a continuous map.

References

• Wolfgang Franz (1967). General Topology. Harrap, 56. 
• J.L. Kelley (1955). General topology. van Nostrand, 94-99. 
• Lynn Arthur Steen; J. Arthur Seebach jr (1978). Counterexamples in Topology. Berlin, New York: Springer-Verlag, 9. ISBN 0-387-90312-7.