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Definition: Let X and Y be topological spaces; let p: X -> Y be a surjective map. The map p is said to be a quotient map provided a subset U of Y is open in Y if and only if p^(-1)(U) is open in X. To say that p is a quotient map is equivalent to saying that p is continuous and p maps saturated open sets of X to open sets of Y (or saturated closed sets of X to closed sets of Y).
Source: Topology (second edition) by James R. Munkres