The fuzzy tychonoff theorem

Abstract

Much of topology can be done in a setting where open sets have “fuzzy boundaries.” To render this precise, the paper first describes cl∞-monoids, which are used to measure the degree of membership of points in sets. Then L- or “fuzzy” sets are defined, and suitable collections of these are called L-topological spaces. A number of examples and results for such spaces are given. Perhaps most interesting is a version of the Tychonoff theorem which gives necessary and sufficient conditions on L for all collections with given cardinality of compact L-spaces to have compact product.