  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
$\omega^*$ is equivalent to finding a countable OHI, extremally disconnected,
zerodimensional space with a remote weak P-point. We also present methods which
allow us to find lonely points in a large subspace of $\omega^*$ and show why
known methods do not allow us to construct them in all of $\omega^*$.
