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One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map (a Tukey quotient) $\phi :P \to Q$ carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients of each other are said to be Tukey equivalent. Let ${\cal D}_{\rm{}} $ be the partially ordered set of Tukey equivalence classes of directed sets of size $ \le {\rm{}}$ . It is shown that ${\cal D}_{\rm{}} $ contains an antichain of size $2^{\rm{}} $ , and so has size $2^{\rm{}} $ . The elements of the antichain are of the form ${\cal K}\left( M \right)$ , the set of compact subsets of a separable metrizable space M, ordered by inclusion. The order structure of such ${\cal K}\left( M \right)$ ’s under Tukey quotients is investigated. Relative Tukey quotients are introduced. Applications are given to function spaces and to the complexity of weakly countably determined Banach spaces and Gul’ko compacta.



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[1]Argyros, S. A., Arvanitakis, A. and Mercourakis, S., Talagrand’s K σδproblem. Topology and its Applications, vol. 155 (2008), no. 15, pp. 17371755.
[2]Arkhangel’skii, A. V.. Topological Function Spaces. Mathematics and its Applications, vol. 78, Kluver Academic Publishers, Netherlands, 1992.
[3]Avilés, A., Weakly countably determined spaces of high complexity. Studia Mathematica, vol. 185 (2008), pp. 291303.
[4]Baars, J., De Groot, J. and Pelant, J., Function Spaces of Completely Metrizable Spaces. Transactions of the American Mathematical Society, vol. 340 (1992), no. 2, pp. 871883.
[5]Christensen, J. P. R., Topology and Borel Structure, North-Holland, Amsterdam-London; American Elsevier, New York, 1974.
[6]Dobrinen, N. and Todorčević, S., Tukey types of ultrafilters. Illinois Journal of Mathematics, vol. 55 (2011), no. 3, pp. 907951.
[7]Dobrinen, N. and Todorčević, S., A new class of Ramsey-classification theorems and their application in the Tukey theory of ultrafilters, Part 1. Transactions of the American Mathematical Society, vol. 366 (2013), no. 3, pp. 16591684.
[8]Fremlin, D. H., Families of compact sets and Tukey’s ordering, Atti del Seminario Matematico e Fisico dell’Università di Modena, vol. 39 (1991), no. 1, pp. 2950.
[9]Fremlin, D. H., The partially ordered sets of measure theory and Tukey’s ordering. Note di Matematica, vol. 11 (1991), pp. 177214.
[10]Fremlin, D. H., Measure Theory, Volume 5, Torres Fremlin, Colchester, 2000.
[11]Husek, M. and van Mill, J. (editors), Recent Progress in General Topology II, North-Holland, Amsterdam, 2002.
[12]Isbell, J. R., Seven cofinal types. Journal of London Mathematical Society (2), vol. 4 (1972), pp. 394416.
[13]Knight, R. W. and McCluskey, A. E., ${\cal P}$(ℝ) ordered by homeomorphic embeddability, does not represent all posets of cardinality $2^{\rm{}} $. Topology and its Applications, vol. 156 (2009), pp. 19431945.
[14]Louveau, A. and Velickovic, B., Analytic ideals and cofinal types. Annals of Pure and Applied Logic, vol. 99 (1999), pp. 171195.
[15]Mamatelashvili, A., Tukey Order on Sets of Compact Subsets of Topological Spaces, PhD thesis,
[16]Marciszewski, W. and Pelant, J., Absolute Borel Sets and Function Spaces. Transactions of the American Mathematical Society, vol. 349 (1997), no. 9, pp. 35853596.
[17]McCluskey, A. E., McMaster, T. B. M., and Watson, W. S.Representing set-inclusion by embeddability (among the subspaces of the real line). Topology and its Applications, vol. 96 (1999), pp. 8992.
[18]Milovich, D., Tukey classes of ultrafilters on ω. Topology Proceedings, vol. 32 (2008), pp. 351362. Spring Topology and Dynamics Conference,
[19]Moore, E. H. and Smith, H. L., A general theory of limits. American Journal of Mathematics, vol. 44 (1922), pp. 102121.
[20]Moore, J. T. and Solecki, S., A G δideal of compact sets strictly above the nowhere dense ideal in the Tukey order. Annals of Pure and Applied Logic, vol. 156 (2008), pp. 270273.
[21]Raghavan, D. and Todorčević, S., Cofinal types of ultrafilters. Annals of Pure and Applied Logic, vol. 163 (2012), pp. 185199.
[22]Solecki, S. and Todorčević, S., Cofinal types of topological directed orders. Annales de l’Institut Fourier, Grenoble, vol. 54 (2004), no. 6, pp. 18771911.
[23]Solecki, S. and Todorčević, S., Avoiding families and Tukey functions on the nowhere-dense ideal. Journal of the Institute of Mathematics of Jussieu, vol. 10 (2011), no. 2, pp. 405435.
[24]Talagrand, M., Espaces de Banach faiblement ${\cal K}$-analytiques. Annals of Mathematics (2), vol. 110 (1979), no. 3, pp. 407438.
[25]Todorčević, S., Directed sets and cofinal types. Transactions of the American Mathemtical Society, vol. 290 (1985), no. 2, pp. 711723.
[26]Tukey, J. W., Convergence and unifomity in topology. Annals of Mathematics Studies, vol. 2, Princeton University Press, Princeton, 1940.
[27]Vasak, L., On a generalization of weakly compactly generated spaces. Studia Mathematica, vol. 70 (1981), pp. 1119.





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