Department of Mathematics and Statistics, Youngstown State University, Youngstown OH 44555-0001 USA, e-mail: zpiotr@math.ysu.edu
Abstract: Based on some earlier findings on Banach Category Theorem for some "nice" $\sigma$-ideals by J. Kaniewski, D. Rose and myself I introduce the $h$ operator ($h$ stands for "heavy points") to refine and generalize kernel constructions of A. H. Stone. Having obtained in this way a generalized Kuratowski's decomposition theorem I prove some characterizations of the domains of functions having "many" points of $h$-continuity. Results of this type lead, in the case of the $\sigma$-ideal of meager sets, to important statements of Abstract Analysis such as Blumberg or Namioka-type theorems.
Keywords: Banach Category Theorem, categorical almost continuity, Blumberg space, separate and joint continuity
Classification (MSC 1991): 54E52, 54A25, 54B15, 54C08
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