The topological structure of fuzzy sets with endograph metric
For a non-degenerate convex subset Y of the it-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 OS(Y) with the topology of endograph metric is homeomorphic to the Hilbert cube Q = [-1, 1](omega) iff Y is compact; and the space K(Y) is homeomorphic to {(x(n)) is an element of Q : sup |x(n)| < 1} iff Y is non-compact and locally compact. (C) 2009 Elsevier B.V. All rights reserved.
fuzzy sets endograph metric the hilbert cubes sigma