Abstract

Kenderov [Continuity-like properties of set-valued mappings, Serdica Bulg. Math. Publ. 9 (1983) 149–160] proved a general result stating that an arbitrary multivalued mapping from a topological space XX to a set YY has some properties resembling continuity at every point of a residual subset X0⊂XX_0\subset X (i.e. its complement X∖X0X\setminus X_0 is of first Baire category). This statement has far-reaching consequences and can be called a ``continuity phenomenon", since it proves and unifies in a general approach several different results in topology and functional analysis, mainly concerning single-valuedness almost everywhere (in the topological sense) of multivalued mappings. In this paper we show that, in the case when XX is a metric space and YY is a compact separable topological space, the set X0X_0 is even σ\sigma-full cone porous (a notion introduced here). It implies that the above (and other) ``generic" results have ``σ\sigma-full cone porous" versions, with unified proofs.

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Keywords

  • Porous sets
  • fully cone porous set
  • generalized monotone mapping
  • submonotone mapping
  • lower almost continuity

Mathematics Subject Classification

54C08, 47H05

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Suggested citation

P. G. Georgiev. “Continuity Phenomenon of Kenderov and Porosity: the Case of Countable Systems.” Journal of Convex Analysis 31 (2024), No. 4, 1139–1150. https://doi.org/10.68381/jca31055

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.