Topology Atlas Document # ppae-28


On lower semicontinuous multifunctions in quasi-uniform and vector spaces

Andrzej Spakowski

Proceedings of the Ninth Prague Topological Symposium (2001) pp. 309-319

Given a cover B of a quasi-uniform space Y we introduce a concept of lower semicontinuity for multifunctions F:X --> 2Y, called B-lsc. In this way, we get a common description of Vietoris-lsc, Hausdorff-lsc, and bounded-Hausdorff-lsc as well. Further, we examine set-theoretical and vector operations on such multifunctions. We also point out that the convex hull of Hausdorff-lsc multifunctions need not to be Hausdorff-lsc except the case where the range space is locally convex.

Mathematics Subject Classification. 54C60 54E15 46A16.
Keywords. Lower semicontinuous multifunctions (set-valued maps), cartesian product, intersection, vector sum, and convex hull of multifunctions..

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Copyright © 2002 Charles University and Topology Atlas. Published April 2002.