Let $S$ be a set and $\vartheta$ be an equivalence relation on $S$. We say that $\vartheta$ is *proper* if there are $x\neq y\in S$ with $(x,y)\in\vartheta$.


Is there an infinite Hausdorff space $(X,\tau)$ such that for every proper equivalence relation $\vartheta$ on $X$ we have $X\not\cong X/\vartheta$?



Infinite Hausdorff space that is not homeomorphic to any proper quotient
Let $X$ be a strongly rigid infinite Hausdorff space. Let $\vartheta$ be an equivalence relation on $X$. Let $q \colon X \to X/\vartheta$ be the quotient map. If there is a homeomorphism $f \colon X/\vartheta \to X$, then $(f \circ q)$ is a continuous map from $X$ to itself; hence constant or the identity. Since $X$ is infinite, we deduce that $\vartheta$ is trivial.


