Journal of Convex Analysis 31 (2024), No. 4, 1139--1150
Copyright Heldermann Verlag 2024
Continuity Phenomenon of Kenderov and Porosity: the Case of Countable Systems
Pando G. Georgiev
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
pandogeorgiev2020@gmail.com
[Abstract-pdf]
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 $X$ to a set $Y$ has some properties resembling continuity at every point of a residual subset $X_0\subset X$ (i.e. its complement $X\setminus X_0$ 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 $X$ is a metric space and $Y$ is a compact separable topological space, the set $X_0$ is even $\sigma$-full cone porous (a notion introduced here). It implies that the above (and other) ``generic" results have ``$\sigma$-full cone porous" versions, with unified proofs.
Keywords: Porous sets, fully cone porous set, generalized monotone mapping, submonotone mapping, lower almost continuity.
MSC: 54C08, 47H05.
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