Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Array convergence of functions of the first Baire class


Author: Helmut Knaust
Journal: Proc. Amer. Math. Soc. 112 (1991), 529-532
MSC: Primary 46E15; Secondary 46B15
DOI: https://doi.org/10.1090/S0002-9939-1991-1057955-9
MathSciNet review: 1057955
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that every array $ (x(i,j):1 \leq i < j < \infty )$ of elements in a pointwise compact subset of the Baire-$ 1$ functions on a Polish space, whose iterated pointwise limit $ {\lim _i}{\lim _j}x(i,j)$ exists, is converging Ramsey-uniformly. An array $ (x{(i,j)_{i < j}})$ in a Hausdorff space $ T$ is said to converge Ramsey-uniformly to some $ x$ in $ T$, if every subsequence of the positive integers has a further subsequence $ ({m_i})$ such that every open neighborhood $ U$ of $ x$ in $ T$ contains all elements $ x({m_i},{m_j})$ with $ i < j$ except for finitely many $ i$.


References [Enhancements On Off] (What's this?)

  • [1] T. K. Boehme and M. Rosenfeld, An example of two compact Hausdorff Fréchet spaces whose product is not Fréchet, J. London Math. Soc. 8 (1974), 339-344. MR 0343242 (49:7986)
  • [2] J. Bourgain, D. Fremlin, and M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1978), 845-886. MR 509077 (80b:54017)
  • [3] E. E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163-165. MR 0349393 (50:1887)
  • [4] K. Kuratowski, Topology, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [5] E. Odell, Applications of Ramsey theorems to Banach space theory, Notes in Banach Spaces (H. E. Lacey, ed.), Univ. Texas Press, Austin, 1980, pp. 379-404. MR 606226 (83g:46018)
  • [6] E. Odell and H. P. Rosenthal, A double dual characterization of separable Banach spaces containing $ {l^1}$, Israel J. Math. 20 (1975), 375-384. MR 0377482 (51:13654)
  • [7] J. D. Pryce, A device of R. J. Whitley's applied to pointwise compactness in spaces of continuous functions, Proc. London Math. Soc. 23 (1971), 532-546. MR 0296670 (45:5729)
  • [8] H. P. Rosenthal, Some remarks concerning unconditional basic sequences, Longhorn Notes 1982-83, The University of Texas, Austin, pp. 15-48. MR 832215
  • [9] J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60-64. MR 0332480 (48:10807)
  • [10] J. Stern, A Ramsey theorem for trees, with an application to Banach spaces, Israel J. Math. 29 (1978), 179-188. MR 0476554 (57:16114)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E15, 46B15

Retrieve articles in all journals with MSC: 46E15, 46B15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1057955-9
Keywords: First Baire class, array convergence, Ramsey theory
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society