The topological structures of spaces of fuzzy n-cell numbers and fuzzy n-ellipsoid numbers

Abstract

As two special types of the general n-dimension fuzzy numbers, fuzzy n-cell numbers and fuzzy n-ellipsoid numbers play an important role in the representations of imprecise multichannel digital information. Denote the families of all n-dimension fuzzy numbers, fuzzy n-cell numbers and fuzzy n-ellipsoid numbers, by K(Rn), L(Rn) and E(Rn), respectively. In this paper, we study the topological structures of the sets L(Rn) and E(Rn) with the topology induced by the supremum metric D∞. Our main result is as follows: for every n>1, there exist homeomorphisms H1,H2:(K(Rn),D∞)→[0,+∞)×ℓ2(c) such that H1(L(Rn),D∞)=H2(E(Rn),D∞)={0}×ℓ2(c), where ℓ2(c) is a non-separable Hilbert space whose weight is the cardinality of the set of all real numbers. As a consequence, the spaces (L(Rn),D∞) and (E(Rn),D∞) are homeomorphic to the Hilbert space ℓ2(c), and hence do not have the fixed point property.