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?



Non-homogenizable topological spaces
This problem is classical and is discussed in $\S 3$ of [this paper](http://www.mathnet.ru/links/6dbb3e53dec0d3d2a368270f9aee4a2e/rm2424.pdf) of Arkhangelski. In particular, in (3.24) he mentions the following 


**Theorem (Okhromeshko, 1983).** Each topological space $X$ is a retract of its free homogeneous space $H(X)$.


The free homogeneous space $H(X)$ in Okhromeshko Theorem is the free group over $X$ endowed with the strongest topology in which for every words $u,v\in H(X)$ the map $X\to H(X)$, $x\mapsto uxv$, is continuous.


The Okhromeshko Theorem implies that each topological space is homogenizable.


