Asymptotic structures of cardinals

Oleksandr Petrenko, Igor V. Protasov, Sergii Slobodianiuk

Abstract

A ballean is a set X endowed with some family F of its subsets, called the balls, in such a way that (X,F)  can be considered as an asymptotic counterpart of a uniform topological space. Given a cardinal k, we define F using a natural order structure on k. We characterize balleans up to coarse equivalence, give the criterions of metrizability and cellularity, calculate the basic cardinal invariant of these balleans. We conclude the paper with discussion of some special ultrafilters on cardinal balleans.

Keywords

cardinal balleans; coarse equivalence; metrizability; cellularity; cardinal invariants; ultrafilter.

Subject classification

54A25; 05A18.

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References

M. Filali, I. V .Protasov, Spread of balleans, Appl. Gen. Topol. 9 (2008), 161-175.

(http://dx.doi.org/10.4995/agt.2008.1796)

T. Jech, Lectures in Set Theory, Lecture Notes in Math. 27 (1971).

K. P. Hart and J. Van Mill, Open problems in $betaomega$, in Open Problems in Topology, J.vanMill, G.M.Reed (Editors), Elsevier Science Publishers, North Holland, 1990, 98-125.

N. Hindman and D. Strauss, Algebra in the Stone-Cech compactification: Theory and Applications, Walter de Gruyter, Berlin, New York, 1998.

(ttp://dx.doi.org/10.1515/9783110809220)

J. Ketonen, On the existence of $P$-points in the Stone-Cech compactification of integers, Fundam. Math. 62 (1976), 91-94.

K. Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland, 1980.

A. R. D.Mathias, $O^#$ and $P$-point problem, Lecture Notes in Mathematics 669 (1978), 375-384.

(http://dx.doi.org/10.1007/BFb0103109)

O. Petrenko and I. V. Protasov, Thin ultrafilters, Notre Dame J. Formal Logic 53 (2012), 79-88.

(http://dx.doi.org/10.1215/00294527-1626536)

O. V. Petrenko and I. V.Protasov, Balleans and $G$-spaces, Ukr. Math. J. 64 (2012), 344-350.

(http://dx.doi.org/10.1007/s11253-012-0653-x)

I. V. Protasov, Resolvability of ball structures, Appl. Gen. Topol. 5 (2004), 191-198.

(http://dx.doi.org/10.4995/agt.2004.1969)

I.V. Protasov, Cellularity and density of balleans, Appl. Gen. Topol. 8 (2007), 283-291.

(http://dx.doi.org/10.4995/agt.2007.1898)

I. V. Protasov, The combinatorial derivation, Appl. Gen. Topol. 14 (2013), 171-178.

(http://dx.doi.org/10.4995/agt.2013.1587)

I. V.Protasov,Extraresolvability of balleans, Comment. Math. Univ. Carolinae 48 (2007), 161-175.

I. V.Protasov, Asymptotically scatterd spaces, preprint (arXiv:1212.0364).

I. V. Protasov and M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., Vol. 12, VNTL Publishers, Lviv, 2007.

J. Roe, Lectures on Coarse Geometry, Amer. Math. Soc., Providence, R.I, 2003.

W. Rudin, Homogenity problems in the theory of Cech compactifications, Duke Math. J. 23 (1956), 409-419.

(http://dx.doi.org/10.1215/S0012-7094-56-02337-7)

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