Topological properties of the spaces of regular fuzzy sets (Q1103222)

For a metric space X a regular fuzzy set on X is defined to be a function \(u: X\to [0,1]\) satisfying the following three conditions: (1) supp\(u=\{x: u(x)>0\}^-\) is compact; (2) u is upper semi-continuous; (3) \(\sup u=1\). For a normed linear space, the additional requirement is \(u(ax_ 1+(1-a)x_ 2)\geq \min (u(x_ 1),u(x_ 2))\) whenever \(0\leq a\leq 1.\) The author studies the set \(F_ 0(X)\) of regular fuzzy sets on a metric space X. It is topologized by identifying it with a subspace \(T_ 0(X)\) of some compact set-valued function space. Some properties of this topology are studied. When X is a reflexive Banach space, \(F_ 0(X)\) is embedded into a locally convex topological vector space and a theory of integration and differentiation has been built up.