Notes on Retracts of Coset Spaces
We study retracts of coset spaces. We prove that in certain spaces the set of points that are contained in a component of dimension less than or equal to $n$, is a closed set. Using our techniques we are able to provide new examples of homogeneous spaces that are not coset spaces. We provide an example of a compact homogeneous space which is not a coset space. We further provide an example of a compact metrizable space which is a retract of a homogeneous compact space, but which is not a retract of a homogeneous metrizable compact space.