Let $X$ be a separable metric space which is *homogeneous*, i.e. for every two points $x,y\in X$ there is a homeomorphism $h$ of $X$ onto itself such that $h(x)=y$.


A compactification of $X$ is a compact metric space which contains a dense homeomorphic copy of $X$.


Does $X$ have a homogeneous compactification?


Examples of homogeneous compactifications include the circle for the real line, the torus for the plane etc.



Do all homogeneous spaces have homogeneous compactifications?
Since you want a connected example:


A surface of infinite genus has no homogeneous compactification.


Indeed first observe a dense locally compact subset has to be open.


So the surface has to be open, and by homogeneity the compactification is a closed surface. But an open subset of a closed surface has (each component of) finite genus.


