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By Klaas Pieter (nl-delfem

Abstract

In the study of the operators preserving disjointness between vector lattices, there are natural and basic questions still unsolved. Along this line, Y. A. Abramovich and A. K. Kitover [Positivity 7 (2003), no. 1-2, 95–97; MR2028371 (2004j:47076)] posed some open problems related with the notions of the so-called d-independence and d-basis in a vector lattice. In particular, they asked the following topological question: Is there a connected compact F-space K admitting a continuous real function f ∈ C(K) which is not essentially constant? Recall that a continuous function f on a compact space K is said to be essentially constant if there is a collection of open subsets of K such that their union is dense in K and f is constant on each of these sets. The purpose of the note under review is to give a positive answer to the above question. For that, the author constructs in a nontrivial way such a compact space K which is in particular a closed subspace of the Stone-Čech remainder S ∗ = βS � S, where S = ω × [0, 1] × [0, 1]. On the other hand, the existence of this space K can also be used to give a negative answer to another of the above-mentioned questions posed by Abramovich and Kitover

Year: 2011
OAI identifier: oai:CiteSeerX.psu:10.1.1.187.4247
Provided by: CiteSeerX
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