Quasi-continuity of multivalued maps with respect to the qualitative topology

Abstract

A subset A of a topological space X is said to be: semi-open, if there exists an open set U such that U c A ~ U [8], semi-closed, if X \ A is semi-open [2, 3]. The union of any family of semi-open sets is semi-open [8]. The union of all semi-open sets which are contained in A is called the semi-interior of A; we denote it by s-Int A [2, 3]. From the definition it immediately follows that each semi-open (semi-closed) set has the Baire property. Let F be a multivalued map which assigns a non-empty subset F(x) of a topological space Y to each point xCX (for simplicity we will write F: X~Y). For any sets AcJ(, B c Y wewilldenote [I]: F(A)= U {F(x): xEA}, F+(B)={xEX:F(x)c cB}, F-(B)={xEX: e(x)NBr A multivalued map F: X ~ Y is said to be: upper (lower) quasi-continuous at a point XoEX, if for each open set V c Y such that F(Xo)c V (resp. F(xo)N V# O) and for each neighbourhood U of x0 there exists an open non-empty set U1cU such that F(x)cV (resp. F(x)NVr for each xC U1 [10, 12[, quasi-continuous at xoEX, if for any open sets V~, V~c Y such that F(xo)C c V1 and F(Xo) N 112 r ~ and for each neighbourhood U of x0 there exists an open non-emptyset U ~ U such that F(x)cV~ and F(x)NV2r for xEU~ [13]. Any single valued map f : X-~Y can be considered as a multivalued map with values {f(x)}. In this case each of the above three definitions gives the definition of quasi-continuity in the sense of Kempisty [6]. In the sequel the symbol E,(F), El(F) and E,(F) will be used to denote the sets of all points at which a multivalued map F is upper quasi-continuous, lower quasicontinuous or quasi-continuous respectively. It follows from the definitions that E,(F) c E u(F) NEt (F); the equality does not hold in general [4]. A multivalued map F is called upper quasi-continuous (lower quasi-continuous, quasi-continuous) if it has this property at each point Equivalently F is upper (lower) quasi-continuous iff for each open set V c Y the set F+(V) (resp. F-(V)) is semiopen. Similarly Fis quasi-continuous ifffor any open sets V~, V2~ Y the set F+(VI) N 0 F-(V2) is semi-open [13]. Let J( be a topological space and let Tq = {U',,,H: U is open, H is of the first category}. Then Tq is a topology on X [5] which sometimes is called the qualitative topology.