A topological space $(X,\tau)$ is said to be *homogeneous* if for $x,y\in X$ there is a homeomorphism $\varphi: X\to X$ such that $\varphi(x) = y$. Let us call a topological space $(X,\tau)$ *homogenizable* if there is a homogeneous space $X_h$ and a continous map $e_X: X \to X_h$ such that for any homogeneous space $Z$ and continous map $f: X\to Z$, there is a continous map $g: X_h\to Z$ such that $f = g\circ e_X$.

What is an example of a non-homogenizable topological space?