Continuity Phenomenon of Kenderov and Porosity: the Case of Countable Systems

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.