The topological structure of fuzzy sets with endograph metric

  • Authors:

  • Zhongqiang Yang;Lili Zhang

  • Affiliations:

  • Department of Mathematics, Shantou University, Shantou, Guangdong 515063, PR China;Department of Mathematics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, PR China

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2009

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Abstract

For a non-degenerate convex subset Y of the n-dimensional Euclidean space R^n, let K(Y) be the family of all fuzzy sets of R^n, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in Y. We show that the space K(Y) with the topology of endograph metric is homeomorphic to the Hilbert cube Q=[-1,1]^@w iff Y is compact; and the space K(Y) is homeomorphic to {(x"n)@?Q:sup|x"n|