Subcontinuity

Abstract

ABSTRACT. We give interesting characterizations using subcontinuity. Let X, Y be topological spaces. We study subcontinuity of multifunctions from X to Y and its relations to local compactness, local total boundedness and upper semicontinuity. If Y is regular, then F is subcontinuous iff F is USCO. A uniform space Y is complete iff for every topological space X and for every net {F a }, F a ⊂ X × Y , of multifunctions subcontinuous at x ∈ X, uniformly convergent to F , F is subcontinuous at x. A Tychonoff space Y isČech-complete (resp. G m -space) iff for every topological space X and every multifunction F ⊂ X × Y the set of points of subcontinuity of F is a G δ -subset (resp. G m -subset) of X. There is a very natural extension of subcontinuity to multifunctions, introduced by Hrycay [22]1 . There is a large number of publications dealing with subcontinuity of either functions or multifunctions. Subcontinuity is often treated as a supplementary property, like before mentioned "upgrade" of closed graph property 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 54C60, 54A25, 54E15. K e y w o r d s: subcontinuous, locally totally bounded, USCO multifunction, Vietoris topology, Hausdorff uniformity,Čech complete space. Both authors were supported by VEGA 2/0047/10. 1 This is the first reference we were able to find. Probably Smithson [34] did it independently in 1975, his definition is cited more frequently.