Affiliations: [a]
School of Mathematics and Computational Science, Wuyi University, Guangdong, China P.R. | [b]
Department of Mathematics, Shantou University, Guangdong, China P.R.
Correspondence:
[*]
Corresponding author. Hanbiao Yang, School of Mathematics and Computational Science, Wuyi University, Jiangmen city, Guangdong, China P.R. E-mail: hongsejulebu@sina.com.
Note: [1] This author was supported by Natural Science Foundation of China (No. 11871379, No. 11971287), Guangdong Natural Science Foundation (No. 2016A030310002), Innovation Project of Department of Education of Guangdong Province (No. 2018KTSCX231, No. 2018GXJK192) and by The PhD Start-up Fund of Wuyi University (No. 2015BS08).
Note: [2] This author was supported by the National Natural Science Foundation of China (No. 11471202).
Abstract: 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.
Keywords: Fuzzy star-shaped numbers, fuzzy numbers, the endograph metric, the Hilbert cube, the pseudoboundary of the Hilbert cube
