A note on strong quasi-continuity of multifunctions

We generalize the notion of strong quasi-continuity for real functions of real variable, given by Grande (Real Anal Exchange 21(1):236–243, 1995/1996), to the case of multifunctions in topological spaces. We introduce a differentiation base in a metric space and show that a strong quasi-continuous multifunction with respect to this differentiation base is almost everywhere continuous.


Introduction
The notion of quasi-continuity, introduced by Kempisty [10] for real functions, has been studied intensively. There are various reasons for the interest in this notion. Perhaps the main ones are its applications in mathematical analysis and topology. There are also some connections to probability and, of course, to other generalized continuity concepts such as cliquishness [3,22], semi-continuity [14], and α-continuity (called also strong quasi-continuity) [18,19].
Kempisty's definition of quasi-continuity was reformulated for general topological spaces by Neubrunn [18] and generalized to the case of multifunctions by Popa [20]. Many new features evolved in this case. Afterwards, quasi-continuity of multifunctions has been widely considered by many authors, we refer to Neubrunn [17,18] [2], Ewert and Neubrunn [4], Holá and Holý [9], and Matejdes [16]. The special case of quasi-continuous functions of two real variables was investigated by Grande in [5,7,8].
As we have alluded earlier, strong quasi-continuity has been considered in the literature first by Noiri [19] for functions and then by Neubrunn for multifunctions [18]; under a different name-continuity with respect to the α-topology of a topological space.
Strong quasi-continuity of real functions was also considered by Grande in [6] but in a different sense. His definition is based on the density topology in the space of real numbers. We generalize this notion to the case of multifunctions (in abstract spaces) and show that a multifunction which is strongly quasi-continuous is almost everywhere continuous.

Preliminaries
We will use standard notation. The sets of positive integers and real numbers will be denoted by N and R, respectively. R n , n ∈ N, will denote the n-dimensional Euclidean space. Let (S, T(S)) and (Z , T(Z )) be topological spaces. For a set A ⊂ S, Cl(A) and Int(A) will denote the closure and the interior of A, respectively. Moreover, if s 0 ∈ S, then B(s 0 ) ⊂ T(S) will denote the family of all neighbourhoods of s 0 .
Kempisty's definition of quasi-continuity of a function f : X → Y was given for the case X = R n and Y = R [10]. For general topological spaces it can be reformulated in the following way [18]: Clearly any continuous function is quasi-continuous. The converse is of course not true. Any left or right continuous function f : R → R is quasi-continuous.
By a multifunction (set-valued map) from S to Z we mean a map which assigns to every point of S a nonempty subset of Z . If is a multifunction from S to Z , we denote it by : S ; Z .
If : S ; Z and G ⊂ Z are given, then we define Note that .
A function f : S → Z may be considered as a multifunction assigning to s ∈ S the singleton { f (s)}. It is clear that in this case for G ⊂ Z we have A multifunction : S ; Z is called upper (resp. lower) semicontinuous at a point ; is called continuous at s 0 ∈ S if it is both upper and lower semicontinuous at s 0 . is called continuous or upper (resp. lower) semicontinuous if it is continuous or upper (resp. lower) semicontinuous at each s ∈ S. Following Neubrunn [18], a multifunction : S ; Z is lower (resp. upper) quasicontinuous at a point s 0 ∈ S if for any set G ∈ T(Z ) such that s 0 ∈ − (G) (resp. s 0 ∈ + (G)) and for any set Note that in the case of single-valued function the notions of lower quasi-continuity and upper quasi-continuity coincide with quasi-continuity.
A multifunction : S ; Z is said to be quasi-continuous at a point s 0 ∈ S if for arbitrary sets G ∈ T(Z ) and H ∈ T(Z ) such that s 0 ∈ − (G) ∩ + (H ) and for every set Clearly a quasi-continuous multifunction is both lower quasi-continuous and upper quasi-continuous. The converse is not true (see [18,Example 1.2.7]). Also, any lower (resp. upper) semicontinuous multifunction is lower (resp. upper) quasi-continuous while the converse is not true. A

Main results
Let us assume that (S, d, M(S), μ) is a measure metric space with metric d, with a σ -finite complete and G δ -regular measure μ defined on a σ -field M(S) containing Borel sets; μ * will denote the outer measure generated by μ, i.e., μ In Let F ⊂ M(S) be a family of sets with nonempty interiors of positive and finite measure μ, the boundaries of which are μ-negligible. Let {I n } n∈N ⊂ F and s ∈ S. We write I n → s to mean that for each n ∈ N, s ∈ Int(I n ) and the diameter of I n tends to zero as n runs to infinity. We assume that for every s ∈ S, there exists a sequence (I n ) n∈N of sets from F such that I n → s. (whenever exist) taken for all the sequences (I n ) n∈N selected from F and converging to s. Replacing supremum by infimum we obtain the lower outer density of A at s ∈ S with respect to F. These densities we will denote by D * u (A, s) and D * l (A, s), respectively. If these densities coincide, then their common value is called the outer density of the set A at s with respect to F and is denoted by D * (A, s). If A ∈ M(S), then the outer density of the set A at s ∈ S with respect to F is called the density of A at s with respect to F and is denoted by D(A, s).
A point s ∈ S is called the density point of a set A ⊂ S with respect to F if there exists a set B ∈ M(S) such that B ⊂ A and the density of B at s with respect to F is equal to 1. We will write D(A, s) = 1.
We will assume that (DP) F has the density property, i.e., every set A ⊂ S satisfies the following condition: μ({s ∈ A : D * l (A, s) < 1}) = 0. By the density property of F, it is clear that

Lemma 3.1 A set A ⊂ S, whose μ-almost every point is its density point with respect to F, is M(S)-measurable.
An M(S)-measurable set is called homogeneous with respect to F if its density with respect to F is 1 at each of its points. The space S can be topologized by taking the homogeneous sets with respect to F as open sets (see [11, p. 232]). This topology we will denote by T D (S). Note that T D (S) is finer than T d (S). The interior of A with respect to T D (S) is denoted by T D -Int(A).
From now on let (S, d, M(S), μ) be a measure metric space with the differentiation basis (F, →) with the density property (DP), and let (Z , T(Z )) be a topological space. Definition 3.2 A multifunction : S ; Z is called strongly lower (resp. upper) quasi-continuous at a point s 0 ∈ S with respect to F if for any set G ∈ T(Z ) such that s 0 ∈ − (G) (resp. s 0 ∈ + (G)) and for any set U ∈ T D (S) containing s 0 there exists a nonempty open set V ⊂ S such that V ∩ U = ∅ and V ∩ U ⊂ − (G) (resp. V ∩ U ⊂ + (G)); is said to be strongly lower (resp. upper) quasi-continuous with respect to F if it is strongly lower (resp. upper) quasi-continuous with respect to F at every point s ∈ S.
Observe that replacing in the above definition the density topology with the topology generated by the metric d, we obtain the notion of lower (resp. upper) quasi-continuity of . Since Clearly a multifunction : S ; Z which is strongly quasi-continuous with respect to F is quasi-continuous. Furthermore, if is strongly quasi-continuous with respect to F, then it is both strongly lower quasi-continuous and strongly upper quasi-continuous with respect to F.
Some connections between the quasi-continuity and the Denjoy property of real functions were considered by Šalát [21]. We establish now more general properties for multifunctions. Definition 3.5 A multifunction : S ; Z is said to have the D − (resp. D + ) property if for any set G ∈ T(Z ) and for any nonempty open set U ⊂ S the set U ∩ − (G) (resp. U ∩ + (G)) is either empty or μ * (U ∩ − (G)) > 0 (resp. μ * (U ∩ + (G)) > 0).

Proposition 3.6
If a multifunction : S ; Z is lower (resp. upper) quasi-continuous, then has the D − (resp. D + ) property.
For a multifunction we denote by D( ), D l ( ) and D u ( ) the sets of all its discontinuity, lower semidiscontinuity and upper semidiscontinuity points, respectively. The following lemma will be useful.
where (V n ) n∈N is a sequence of all finite unions of sets from B.

Theorem 3.8 If the space (Z , T(Z )) is regular and second countable, a multifunction
: S ; Z is strongly lower quasi-continuous with respect to F and it has the D + property, then μ(D l ( )) = 0.
Similar proof works for a dual theorem.