Quotient topology

In general topology, the quotient 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.