$I^{X}$, the hyperspace of fuzzy sets, a natural nontopological fuzzy topological space

Abstract:

Let $X$ be a uniform topological space, then on the family ${I^X}$ (resp. $\Phi (X)$) of all nonzero functions (resp. nonzero uppersemicontinuous functions) from $X$ to the unit interval $I$, a fuzzy uniform topology is constructed such that ${2^X}$ (resp. $\mathcal {F}(X)$), the family of all nonvoid (resp. nonvoid closed) subsets of $X$ equipped with the Hausdorff-Bourbaki structure is isomorphically injected in ${I^X}$ (resp. $\Phi (X)$). The main result of this paper is a complete description of convergence in ${I^X}$, by means of a notion of degree of incidence of members of ${I^X}$. Immediate consequences are that first it can be shown that this notion of convergence refines some particular useful notions of convergence of fuzzy sets used in applications, and that second it follows from its construction and properties that for each ordinary uniform topological space $X$ there exists a natural nontopological fuzzy uniform topology on ${I^X}$.