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Define the map <math>q : X \to X / {\sim}</math> as above (by <math>q(x) := [x]</math>) and give <math>X / \sim</math> the quotient topology induced by <math>q</math> (which makes <math>q</math> a quotient map). These maps are related by: <math display=block>\pi = \hat{\pi} \circ q \quad \text{ and } \quad q = \hat{\pi}^{-1} \circ \pi.</math>
From this and the fact that <math>q : X \to X / \sim</math> is a quotient map, it follows that <math>\pi : X \to Y</math> is continuous if and only if this is true of <math>\hat{\pi} : X / \sim \;\;\to\; Y.</math> Furthermore, <math>\pi : X \to Y</math> is a quotient map if and only if <math>\hat{\pi} : X / \sim \;\;\to\; Y</math> is a [[homeomorphism]] (or equivalently, if and only if both <math>\hat{\pi}</math> and its inverse are continuous).
'''Categorical definition'''
A map <math>f : X \to Y</math> is a '''quotient map''' if it is the [[coequalizer]] of its own kernel pair.
=== Related definitions ===
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