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 X ˇ 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 = g p ) . 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.