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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

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