On a question of Mauldin and Ulam concerning homeomorphisms (Q719517)

Let \(E\), \(F\) be topological spaces. Given \(T:E\to F\), we say that \(T\) is an MU-map if, whenever \(A,B\subseteq E\) are homeomorphic then so are \(T(A), T(B)\subseteq F\). The following are the main results of the paper. Theorem. Let \(T:E\to F\) be a continuous onto MU-map. Then, \(T\) is a homeomorphism, provided one of the conditions below holds: i) \(F\) is weakly discretely generated (for each non-closed \(A\subseteq F\) there exists a discrete \(D\subseteq A\) such that \(\text{cl}(D)\setminus A\neq \emptyset\)), ii) \(E\) is a boundedly compact metric space and \(F\) is a Hausdorff space.