On the Cellularity of βX - X

Abstract

For a topological space X, let c(X) denote the cellularity of X, and let k(X) denote the least cardinal of a cobase for the compact subsets of X. It is shown that, if X is a completely regular Hausdorff space, c(,/X X) 2k(X), then f3X-X contains a discrete C*-embedded subspace of cardinality k(X)+. 1. Preliminaries. Our topological notation and terminology follows that of [6]. The Stone-tech compactification of a completely regular Hausdorff space X is denoted by fX, and the cardinality of a set S is denoted by ISI. The discrete space of cardinality a is denoted by D(a). The cardinal 2' is denoted by exp a. Recall that the cellularity of a topological space Y, denoted by c(Y), is defined by c(Y) = sup{IGI: 9 is a family of pairwise disjoint nonempty open subsets of Y}. A family 'X of compact subsets of Y is called a cobase for the compact subsets of Y, if every compact subset of Y is contained in a member of SC We denote by k(Y) the least cardinality of a cobase for the compact subsets of Y. The Lindelof number of Y, denoted by L(Y), is the least cardinal a such that every open cover of Y has a subcover of cardinality L(Y) for each noncompact space Y. Our principal reference for cardinal invariants of topological spaces is [8]. All hypothesized spaces in this paper are assumed to be completely regular and Hausdorff. 2. The cellularity of /8X X. Disjoint open subsets of /3X X are considered in [3], where the following theorem is established (see 3.3 in [3]). 2.1. THEOREM (COMFORT-GORDON). Let X be a completely regular space and let m be a cardinal number. Then the following assertions are equivalent: (i) the space 8X X admits a collection of m pairwise disjoint, nonempty open subsets; (ii) the space X admits a collection 6& of cozero sets such that 1Q11 = m and such that each member of Gi contains a noncompact zero-set, and, if U, V are distinct members of G1, then U n V has compact closure in X. Our estimate for the cellularity of 8iX X uses the Comfort-Gordon result Presented to the Society, February 28, 1975; received by the editors February 25, 1975. AMS (MOS) subject classifications (1970). Primary 54A25, 54D35; Secondary 54D20.