A special fuzzy star-shaped numbers space with endograph metric

In this paper, for a non-degenerate convex set Y in Rn containing 0, two special function spaces S0 (Y) and E0 (Y) which consist of all fuzzy star-shaped numbers and of all fuzzy numbers in Rn with respect to 0 and their supports being included in Y with the endograph metric D are investigated. Some conclusions and methods in topology are used to discuss the topological structure of (S0 (Y) , D) and the pair ((S0 (Y) , D) , (E0 (Y) , D)). The main results are as follows: 1. The space (S0 (Y) , D) is homeomorphic to the Hilbert cube Q = [-1, 1] N if and only if S0 (Y) is compact if and only if Y is compact. 2. There exists a homeomorphism h : (S0 (Y) , D) → Q such that h (E0 (Y)) = {1} × [-1, 1] N\{1} if Y is compact but not a segment. 3. The space (S0 (Y) , D) homeomorphic to the pseudoboundary of the Hilbert cube if and only if S0 (Y) is non-compact and σ-compact if and only if Y is non-compact and locally compact.