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Minmax bornologies

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Abstract

A bornology ℬ on a set X is called minmax, if the smallest and largest coarse structures on X compatible with ℬ coincide. We prove that ℬ is minmax, if and only if the family ℬ# = {p ∈ βX : {X\B : B ∈ ℬ} ⊂ p} consists of ultrafilters which are pairwise non-isomorphic via ℬ-preserving bijections of X. In addition, we construct a minmax bornology ℬ on 𝜔 such that the set ℬ# is infinite. We deduce this result from the existence of a closed infinite subset in 𝛽𝜔 that consists of pairwise non-isomorphic ultrafilters.

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Authors and Affiliations

  1. Ivan Franko National University of Lviv, Lviv, Ukraine

    Taras Banakh

  2. Faculty of Computer Science and Cybernetics, Kyiv University, Kyiv, Ukraine

    Igor Protasov

Authors
  1. Taras Banakh
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  2. Igor Protasov
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Correspondence to Taras Banakh.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 16, No. 3, pp. 496–502 October–December, 2019.

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Banakh, T., Protasov, I. Minmax bornologies. J Math Sci 246, 617–621 (2020). https://doi.org/10.1007/s10958-020-04767-4

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  • DOI: https://doi.org/10.1007/s10958-020-04767-4

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