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 $$\bar F$$ 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 is Čech-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.