How many miles to βω?—Approximating βω by metric-dependent compactifications

Covering a bounded set of functions by an increasing chain of slaloms

“…In this section, we give an answer to a question which was posed by Kada, Tomoyasu and Yoshinobu [6]. We refer the reader to [6] for undefined topological notions.…”

“…Define H 2 ∈ ω ω by letting H 2 (n) = 2 2 (n 4 ) for all n. The following lemma is obtained as a corollary of the proof of [6,Theorem 6.11].…”

“…In Section 3, we give an application of the lemma presented in Section 2 to another problem in topology. We answer a question on approximations to the Stone-Čech compactification of ω by Higson compactifications of ω, which was posed by Kada, Tomoyasu and Yoshinobu [6].…”

“…In the paper[6], the authors state "If CH holds in a ground model V , and a proper forcing notion P has the Laver property, then l = ℵ 1 holds in the model V P ". But it is inaccurate, since we do not see the values of l h for functions h ∈ V P which are not bounded by any function from V .…”

Approximation theorems for compactifications

“…The paper [6] gave the minimal cardinality to approximate βX by Smirnov compactifications in the case of X = [0, 1) or ω.…”