Abstract
We prove that the number of near-coherence classes of non-principal ultrafilters on the natural numbers is either finite or \( 2^\mathfrak{c} \). Moreover, in the latter case the Stone-Čech compactification βω of ω contains a closed subset C consisting of \( (\mathfrak{r} < \mathfrak{u}) \) pairwise non-nearly-coherent ultrafilters. We obtain some additional information about such closed sets under certain assumptions involving the cardinal characteristics \( (\mathfrak{r} < \mathfrak{u}) \) and \( 2^\mathfrak{c} \).
Applying our main result to the Stone-Čech remainder βℝ+−ℝ+ of the half-line ℝ+ = [0,∞) we obtain that the number of composants of βℝ+ − ℝ+ is either finite or \( 2^\mathfrak{c} \).
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Banakh, T., Blass, A. (2006). The Number of Near-Coherence Classes of Ultrafilters is Either Finite or \( 2^\mathfrak{c} \) . In: Bagaria, J., Todorcevic, S. (eds) Set Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7692-9_8
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