A complete characterization of the level spaces of R(I) and I(I) (Q1088181)

The author applies Stone's \(T_ 0\)-modification of a topological space to obtain a useful description of the level topologies for the fuzzy real numbers and the fuzzy unit interval. There are exactly three non- homeomorphic level spaces for the fuzzy reals and four for the interval. The \(T_ 0\)-modifications of the level spaces can be identified as subspaces of the extended plane with the product of the left-order and right-order topologies. As an application the author determines which level spaces have or do not have each of 15 basic topological properties.