Equivalent conditions to a map being a quotient map

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Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces and let [ilmath]f:X\rightarrow Y[/ilmath] be a map. Then:

• [ilmath]f[/ilmath] is a quotient map

if and only if

• It does either one, or both, of the following:
  1. [ilmath]f[/ilmath] takes saturated open sets to open sets
  2. [ilmath]f[/ilmath] takes saturated closed sets to closed sets

Proof

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Unfortunately left as an exercise