Krdytkyu

Let $X$ and $Y$ be homeomorphic topological spaces, connected by the homeomorphism $f : X rightarrow Y$. Let $sim$ be an equivalence relation on $X$ and $approx$ be an equivalence relation on $Y$.

I would think that there would be some structure that I could place on the equivalence relations $sim$ and $approx$ that would allow me to construct, using the homeomorphism $f$, a homeomorphism $bar{f}$ from $X/sim$ to $Y/approx$. I suspect that that relationship is $a sim b leftrightarrow f(a) approx f(b), forall a, b in X$.

That is, I suspect that "equivalent" quotients of homeomorphic spaces are homeomorphic, but I don't know exactly how to formulate this or prove it.

(This result, by the way, to me, seems to hinge on the existence of a "reverse" universal quotient property: juts as functions $f : X/sim rightarrow Y$ correspond uniquely to a subset of functions $f : X rightarrow Y$, I suspect that functions $f : X rightarrow Y/approx$ correspond uniquely to some subset of functions $f : X rightarrow Y$, but I don't know how this works either.)

Your suspicion is correct. If $X$ and $Y$ are topological spaces with equivalence relations $sim$ and $approx$ and $f:Xto Y$ is a homeomorphism such that $xsim y$ if and only if $f(x)approx f(y)$, then $f$ passes to the quotient as $X/_sim cong Y/_approx$.

Proof goes roughly like this: define $widetilde{f}([x]_sim)doteq [f(x)]_approx$. We have that $widetilde{f}$ is well-defined and injective because of the condition relating $sim$ and $approx$, while surjectivity follows from the surjectivity of $f$. Continuity of $widetilde{f}$ and $widetilde{f}^{-1}$ follow from the universal property of the quotient topologies applied to the relations $widetilde{f}circpi_{sim}=pi_{approx}circ f$ and $widetilde{f}^{-1}circ pi_{approx}=pi_simcirc f^{-1}$ with the continuity of $f$ and $f^{-1}$.

I think you are correct. Here is a sketch of the argument:

$phi:=pi_Ycirc f$ is a continuous surjection: $Xto Y/approx$ and it induces a bijection: $X^*=left { phi^{-1}([y]): [y]in Y/approx right }overset{overlinephi}{rightarrow} Y/approx , $ which is a homeomorphism if and only if $phi$ is a quotient map.

As $phi $ is indeed a quotient map, the induced map is a homeomorphism. So, $X/sim$ and $Y/approx$ will be homoeomorphic if and only if $X^*$ and $X/sim$ are.

This means that the map $[x]mapsto f^{-1}circ pi^{-1}_Y([y])$ must be a homeomorphism. That is, $f([x])mapsto pi^{-1}_Y([y])$ must be a homeomorphism.

But the above map is well-defined if and only if $xsim x'Rightarrow f(x)approx f(x'),$ and in this case, it is obviously continuous, and it should be easy to check that it is bijective with continuous inverse.