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Transactions of the American Mathematical Society

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


Author: R. Lowen
Journal: Trans. Amer. Math. Soc. 278 (1983), 547-564
MSC: Primary 54A40; Secondary 03E72, 54B20
MathSciNet review: 701510
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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}$.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0701510-4
Keywords: Hyperspace, fuzzy uniform topology, Hausdorff-Bourbaki uniformity, convergence
Article copyright: © Copyright 1983 American Mathematical Society