A Basically Disconnected Normal Space Φ With |βΦ – Φ| = 1

Abstract

All spaces considered are completely regular. C*(X) denotes the set of bounded continuous real-valued functions in X. A subspace S of X is called C*-embedded in X if for every f ∈ C*(S) there is ϕ ∈ C*(X) with ϕ⥤S = f.A space X is called almost compact if |βX – X| ≦ 1; basically disconnected if every cozero-set has open closure; extremally disconnected if every open set has open closure; an F-space if every cozero-set is C*-embedded; small if |C*(X)| = 2ω; and weakly Lindelöf if every open cover has a subfamily with and ⋃ dense. A point p of a space X is called a P-point of X if every Gδ-set in X which contains p is a neighborhood of p.ω(X) denotes the weight of X.