Nonmetrizable conformally invariant compactifications of a planar region (Q1084229)

Let Q be a region in the complex plane, let \(\beta\) Q denote the usual Stone-Cech compactification of Q, let \(b_{H^{\infty}}Q\) denote the maximal ideal space of Q, let \(b_ WQ\) denote the Wiener compactification of Q, and let \(b_{\rho}Q\) denote the so called natural compactification of the matrix space Q relative to the matrix \(\rho\). \((b_{\rho}Q\) is the compactification corresponding to the subordination for a closed set F and an open set H defined by \(F<H\) if and only if \(\rho (F,Q-H)>0.)\) The author proves a number of results about these compactifications, giving some characterizations for some of them. In particular, it is proved that if Q is simply connected then, in the lattice of all compactifications of Q, both \(b_{H^{\infty}}Q<b_ WQ<\beta Q\) and \(b_{H^{\infty}}Q<b_{\rho \Gamma}Q<\beta Q\), where \(b_{\rho \Gamma}Q\) is the natural compactification of Q with respect to the hyperbolic metric \(\rho\Gamma\). In addition, it is shown that \(b_ WQ\) and \(b_{\rho \Gamma}Q\) are not comparable in this lattice.