# Quotient topology

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Let ${\displaystyle (X,{\mathcal {T}})}$ 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 ${\displaystyle U}$ of ${\displaystyle Y}$ such that the pre-image ${\displaystyle q^{-1}(U)=\{x\in X\mid q(x)\in U\}\in {\mathcal {T}}_{X}}$.