I couldn't found an english traslation for what I am calling "totally compatible function". I am doing a literal traslation here of what I was taught in my topology lecture, but the definition is the following:
Let f be a function $f:X \rightarrow Y$, and $R$ an equivalence relation. f is said to be compatible with $R$ if $xRy$ implies $f(x)=f(y)$; $f$ is said totally compatible if the implication is in both ways. This allow to define a function $f_R: X/R \rightarrow Y$, so that $f=f_R \circ \Pi $, where $\Pi$ is the projection to the quotient space $\Pi: X \rightarrow X/R$,
Then there is a theorem used to prove homeomorphism between spaces, by passing to the quotient:
Let $f: X \rightarrow Y$ be compatible with $R$, then
1) $f_R$ is surjective iff $f$ is surjective
2)$f_R$ is injective iff $f$ is totally compatible
3)$f:(X,\tau)\rightarrow (Y,\tau')$ is continuous iff $f_R$ is continuous
4)$f$ maps saturated open sets to open sets iff $f_R$ is an open function
If someone has seen this before, I'd like to know how you call it in English and if you know of any textbook where I can find it