Remainders of extremally disconnected spaces and related objects

In this article, we study remainders of extremally disconnected spaces and extremally disconnected remainders. Extremally disconnected spaces satisfying certain homogeneity type conditions are at the center of it. It is established that the absolute X of a non-discrete separable metrizable space M not only is never homogeneous, but never has a homogeneous extension (Theorem 3.2). This extends a classic theorem of Z. Frolik on nonhomogeneity of any extremally disconnected compactum [17], [18]. It is proved that if X is an extremally disconnected space with a homogeneous extension, then, for any compactification bX of X, the remainder Y=bX∖X is countably compact (Theorem 3.7). Among other new results are Theorem 4.6 on remainders of Banach spaces, Theorem 4.8 on remainders of normal topological groups, and Theorem 2.9 on remainders of k-trivial spaces.