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

to a set

Y Y

has some properties resembling continuity at every point of a residual subset

X_0\subset X X 0 ⊂ X

(i.e. its complement

X\setminus X_0 X ∖ 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 X

is a metric space and

Y Y

is a compact separable topological space, the set

X_0 X 0

is even

\sigma σ

-full cone porous (a notion introduced here). It implies that the above (and other) ``generic" results have ``

\sigma σ

-full cone porous" versions, with unified proofs.