THE ALPHA-BOUNDIFICATION OF ALPHA

Abstract

A space X is < alpha-bounded if for all A subset-or-equal-to X with ]A] < alpha cl(x) A is compact. Let B (alpha) be the smallest < alpha-bounded subspace of beta (alpha) containing alpha. It is shown that the following properties are equivalent: (a) alpha is a singular cardinal

(b) B(alpha) is not locally compact

(c) B(alpha) is alpha-pseudocompact

(d) B(alpha) is initially alpha-compact. Define B0(alpha) = alpha and B(n)(alpha) = {cl(beta)(alpha) A: A subset-or-equal-to (alpha), \A\ < alpha} for 0 < n < omega. We also prove that B2(alpha) not-equal B3(alpha) when omega = cf(alpha) < alpha. Finally, we calculate the cardinality of B(alpha) and prove that, for every singular cardinal alpha, \B(alpha)\ = \B(alpha)\alpha = \N(alpha)\cf(alpha) where N(alpha) = {p is-an-element-of beta(alpha): there is A is-an-element-of p with \A\ < alpha}.