A natural topology for fuzzy numbers. (English) Zbl 1020.54005

A generalized fuzzy number is an upper semicontinuous function \(\mu: \overline{\mathbb{R}}\to [0,1]\) which attains the value 1 and is such that \(\mu(t)\geq \max\{\mu(s), \mu(r)\}\) whenever \(s\leq t\leq r\). A bijective correspondence between the set of generalized fuzzy numbers and the product \(\Delta\times\Delta\) (\(\Delta\) is the family of all distribution functions on \(\overline{\mathbb{R}}\)) is established. The set of generalized fuzzy numbers is then topologized by a kind of weak convergence (based on the weak convergence on \(\Delta\)) which turns it into a compact metric space whose topology is shown to be the interval topology induced by the pointwise ordering. A relationship between these fuzzy numbers and the \([0,1]\)-valued version of Hutton’s unit interval is discussed.


54A40 Fuzzy topology
03E72 Theory of fuzzy sets, etc.
26E50 Fuzzy real analysis
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