The topological structure of the set of fuzzy numbers with Lp metric

Abstract

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rn, let K(Y) be the family of all fuzzy sets in Rn, which are upper-semicontinuous, fuzzy convex, normal and with compact supports included in Y. We show that the space K(Y) with the topology of the Lp metric with p≥1, that is, for f,g∈K(Y),Lp(f,g)=(∫01(dH([f]α,[g]α))pdα)1p, is homeomorphic to the Hilbert cube Q=[−1,1]ω if and only if Y is compact; and the space K(Y) is homeomorphic to the pseudoboundary B(Q)=Q∖(−1,1)ω of the Hilbert cube Q=[−1,1]ω if and only if Y is non-compact and locally compact.