Fuzzy Sets Over the Poset I

The author studies fuzzy sets over the poset I = [0, 1] with the usualorder. These form a canonical example of fuzzy sets over a poset discussed in (Tiryaki, ˙I. U. and Brown, L. M. Plain textures and fuzzysets via posets, preprint). Characterizations of these so called “softfuzzy sets” are obtained, and soft fuzzy sets are shown to have a richermathematical theory than classical I-fuzzy sets. In particular soft fuzzypoints behave like the points of crisp set theory with respect to join, andmoreover there exists a Lowen type functor from Top to the constructSF-Top that preserves both separation and compactness.