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?
This problem is classical and is discussed in $\S 3$ of this paper 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.
