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A quasi-order on continuous functions

  • Raphaël Carroy (a1)

We define a quasi-order on Borel functions from a zero-dimensional Polish space into another that both refines the order induced by the Baire hierarchy of functions and generalises the embeddability order on Borel sets. We study the properties of this quasi-order on continuous functions, and we prove that the closed subsets of a zero-dimensional Polish space are well-quasi-ordered by bi-continuous embeddability.

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[1]Duparc, J., Wadge hierarchy and Veblen hierarchy Part I: Borel sets of finite rank, this Journal. vol. 66 (2001), no. 1. pp. 5686.
[2]Fraïssé, R., Sur la comparaison des types d'ordres. Comptes Rendus Mathématique Académie des Sciences. Paris, vol. 226 (1948). pp. 13301331.
[3]Hertling, P., Unstetigkeitsgrade von Funktionen in der effektiven Analysis. Ph.D. thesis. Fern Universität Hagen, 11 1996.
[4]Hertling, P. and Weihrauch, K., On the topological classification of degeneracies. Fachbereich Informatik, Fern Universität Hagen, 1994.
[5]Kechris, A. S., Classical descriptive set theory. Springer Verlag, New York, 1994.
[6]Laver, R., On Fraïssé's order type conjecture. The Annals of Mathematics, vol. 93 (1971). no. 1. pp. 89111.
[7]Laver, R., Better-quasi-orderings and a class of trees. Studies in foundations and combinatorics (Rota, Gian-Carlo, editor), vol. 1, 1978, pp. 3148.
[8]Louveau, A. and Saint-Raymond, J., On the quasi-ordering of Borel linear orders under embeddability, this Journal, (1990), pp. 537560.
[9]Mansfield, R. and Weitkamp, G., Recursive aspects of descriptive set theory, ch. Bqo-theory and Fraïssé's conjecture by Simpson, S. G., pp. 124–138, pp. 124138.
[10]Marcone, A., Wqo and bqo theory in subsystems of second order arithmetic. Reverse mathematics, vol. 21 (2001), pp. 303330.
[11]Moschovakis, Y. N.. Descriptive set theory, American Mathematical Society, 2009.
[12]Nash-Williams, C. S. J. A., On well-quasi-ordering infinite trees. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 61 (1965), p. 697.
[13]Pouzet, M. and Sauer, N., From well-quasi-ordered sets to better-quasi-ordered sets. The Electronic Journal of Combinatorics, vol. 13 (2006), no. 1.
[14]Solecki, S., Decomposing Borel sets and functions and the structure of Baire class 1 functions. Journal of the American Mathematical Society, vol. 11 (1998), pp. 521550.
[15]van Engelen, F., Miller, A.W., and Steel, J., Rigid Borel sets and better quasi order theory. Contemporary mathematics, vol. 65, 1987.
[16]Van Wesep, R., Wadge degrees and descriptive set theory, Cabal seminar 76–77, Springer, 1978. pp. 151170.
[17]Weihrauch, K., The TTE Interpretation of three hierarchies of omniscience principles, Fern Universität. 1992.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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