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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
MathSciNet review: 1057955
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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$.

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Keywords: First Baire class, array convergence, Ramsey theory
Article copyright: © Copyright 1991 American Mathematical Society