Weak P-points in coronas of G-spaces

Let G be a group, X be a discrete G-space, X⁎ be the remainder of the Stone–Čech compactification of X. The corona of X is a factor-space of X⁎ by the smallest by inclusion, closed in X⁎×X⁎ equivalence on X⁎ containing the orbit equivalence ∼ (p∼q⇔∃g∈G:q=gp). For a countable group G and a countable G-space X we prove that the corona of X contains a weak P-point and a P-point provided that there exists a P-point in ω⁎. Then we transfer this statement to the Higson coronas of a proper metric spaces of bounded geometry.