# Characteristic property of the quotient topology

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AKA: the Characteristic property of the quotient topology which redirects here

## Statement

 In this commutative diagram[ilmath]f[/ilmath] is continuous[ilmath]\iff[/ilmath][ilmath]f\circ q[/ilmath] is continuous [ilmath]\xymatrix{ X \ar[d]_{q} \ar[dr]^{f\circ q} & \\ Y \ar[r]_f & Z }[/ilmath]
Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces and let [ilmath]q:X\rightarrow Y[/ilmath] be a quotient map. Then[1]:
• For any topological space, [ilmath](Z,\mathcal{ H })[/ilmath] a map, [ilmath]f:Y\rightarrow Z[/ilmath] is continuous if and only if the composite map, [ilmath]f\circ q[/ilmath], is continuous