Lonely points in

Inspired by the work of J. van Mill we define a new topological type—lonely points. We show that the question of whether these points exist in ω ∗ is equivalent to finding a countable OHI, extremally disconnected, zero-dimensional space with a remote weak P-point. We also present methods which allow us to find lonely points in a large subspace of ω ∗ and show why known methods do not allow us to construct them in all of ω ∗ .