Array convergence of functions of the first Baire class
Author:
Helmut Knaust
Journal:
Proc. Amer. Math. Soc. 112 (1991), 529532
MSC:
Primary 46E15; Secondary 46B15
MathSciNet review:
1057955
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We show that every array of elements in a pointwise compact subset of the Baire functions on a Polish space, whose iterated pointwise limit exists, is converging Ramseyuniformly. An array in a Hausdorff space is said to converge Ramseyuniformly to some in , if every subsequence of the positive integers has a further subsequence such that every open neighborhood of in contains all elements with except for finitely many .
 [1]
T.
K. Boehme and M.
Rosenfeld, An example of two compact Hausdorff Fréchet
spaces whose product is not Fréchet, J. London Math. Soc. (2)
8 (1974), 339–344. MR 0343242
(49 #7986)
 [2]
J.
Bourgain, D.
H. Fremlin, and M.
Talagrand, Pointwise compact sets of Bairemeasurable
functions, Amer. J. Math. 100 (1978), no. 4,
845–886. MR
509077 (80b:54017), http://dx.doi.org/10.2307/2373913
 [3]
Erik
Ellentuck, A new proof that analytic sets are Ramsey, J.
Symbolic Logic 39 (1974), 163–165. MR 0349393
(50 #1887)
 [4]
K.
Kuratowski, Topology. Vol. I, New edition, revised and
augmented. Translated from the French by J. Jaworowski, Academic Press, New
YorkLondon; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR 0217751
(36 #840)
 [5]
E.
Odell, Applications of Ramsey theorems to Banach space theory,
Notes in Banach spaces, Univ. Texas Press, Austin, Tex., 1980,
pp. 379–404. MR 606226
(83g:46018)
 [6]
E.
Odell and H.
P. Rosenthal, A doubledual characterization of separable Banach
spaces containing 𝑙¹, Israel J. Math. 20
(1975), no. 34, 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.
(3) 23 (1971), 532–546. MR 0296670
(45 #5729)
 [8]
Haskell
P. Rosenthal, Some remarks concerning unconditional basic
sequences, Texas functional analysis seminar 1982–1983 (Austin,
Tex.), Longhorn Notes, Univ. Texas Press, Austin, TX, 1983,
pp. 15–47. MR
832215
 [9]
Jack
Silver, Every analytic set is Ramsey, J. Symbolic Logic
35 (1970), 60–64. MR 0332480
(48 #10807)
 [10]
Jacques
Stern, A Ramsey theorem for trees, with an application to Banach
spaces, Israel J. Math. 29 (1978), no. 23,
179–188. MR 0476554
(57 #16114)
 [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), 339344. MR 0343242 (49:7986)
 [2]
 J. Bourgain, D. Fremlin, and M. Talagrand, Pointwise compact sets of Bairemeasurable functions, Amer. J. Math. 100 (1978), 845886. MR 509077 (80b:54017)
 [3]
 E. E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163165. 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. 379404. MR 606226 (83g:46018)
 [6]
 E. Odell and H. P. Rosenthal, A double dual characterization of separable Banach spaces containing , Israel J. Math. 20 (1975), 375384. 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), 532546. MR 0296670 (45:5729)
 [8]
 H. P. Rosenthal, Some remarks concerning unconditional basic sequences, Longhorn Notes 198283, The University of Texas, Austin, pp. 1548. MR 832215
 [9]
 J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 6064. MR 0332480 (48:10807)
 [10]
 J. Stern, A Ramsey theorem for trees, with an application to Banach spaces, Israel J. Math. 29 (1978), 179188. 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:
http://dx.doi.org/10.1090/S00029939199110579559
PII:
S 00029939(1991)10579559
Keywords:
First Baire class,
array convergence,
Ramsey theory
Article copyright:
© Copyright 1991
American Mathematical Society
