Openness properties of real multifunctions on some connected spaces (Q1262567)

The announced purpose of this paper is to extend the results of \textit{B. Ricceri} and \textit{A. Villani} [Rend. Mat. Appl., VII. Ser. 2, 679-687 (1982; Zbl 0524.54008)] and \textit{K. Omiljanowski} [Boll. Unione Mat. Ital, VII. Ser., B1, 649-661 (1987)] to the class of semicontinuous real- valued multifunctions having connected values. Some properties of completely connected spaces are developed as well as a characterization of completely connected spaces. Two new connectivity properties \((C_ 1)\) and \((C_ 2)\) are defined and related to existing connectivity properties. Certain relations are investigated between properties of a given space X of a connectivity character such as connectedness, complete connectedness, properties \((C_ 1)\) and \((C_ 2)\) or local connectedness and the behavior of real multifunctions (either lower or upper semicontinuous) with connected values or of real single-valued continuous functions defined on X and concerning their openness properties such as almost openness, pseudo-almost openness or inductive openness. For example the following characterization of certain connected and locally compact metric spaces is obtained: Let X be a connected and locally compact metric space. Then the following are equivalent: (i) each continuous real function on X is pseudo-almost open; (ii) each lower semicontinuous real multifunction with connected values on X is pseudo- almost open; (iii) each upper semicontinuous real multifunction with connected values on X is pseudo-almost open; (iv) X has property \((C_ 1)\). Further developments are made concerning almost openness of certain multifunctions and local connectedness of the domain space and inductive openness of some real multifunctions.