Equivalent conditions to a map being a quotient map

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Review and demote

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[1]:

• 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