G-1 - Extremally Disconnected Spaces

Abstract

A space is called extremally disconnected (ED) if it is regular and the closure of every open set is open. ED space is zero-dimensional and Tychonoff. But it is no ordinary space. It can never be metrizable unless discrete. It contains no convergent sequence of distinct points. Extremal disconnectedness can be characterized in several useful ways. For every Tychonoff space X: (1) X is ED; (2) every pair of disjoint open sets of X have disjoint closures; (3) for every open sets U, V of X, Cl U ∩ Cl V is equal to Cl (U ∩V); (4) every open subset of X is C∗-embedded, and (5) every dense subset of Xis C∗-embedded. By definition, X is ED if and only if (iff) its Stone–Čech compactification βX is ED. Thus βλ is ED for any infinite cardinal λ. If X is ED then every countable subset is C∗-embedded and hence X contains no convergent sequence of distinct points. Every infinite compact ED space contains a copy of βω. If X is a compact ED space and has cellularity greater than λ for an infinite cardinal λ, then as every open subset of X is C∗-embedded, X contains a copy of βλ.