Quotient topology

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

Let 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 of such that the pre-image .

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.