Countable dense homogeneity property of connected compact manifolds

Let X be a connected compact manifold and H(X) denote the set of all homeomorphisms from X onto itself. In this paper we provide another proof for the known result that every connected compact manifolds are n−homogeneous and thereby countable dense homogeneous (CDH). We also show that every compact manifold can be expressed as a disjoint union of countable dense subsets. Additionally, we prove that for every h ∈ H(X), ∃ a partition of X in to countable dense subsets on which it is invariant. Mainly, we discuss about the denseness of HA(X) = { h ∈ H(X) | h : A → A is a homeomorphism } in H(X) whenever A is a subset of X. Finally, we prove that if f is a contraction function on X, then the set of all homeomorphisms that commutes with f is a nowhere dense subgroup of H(X).