BOUNDEDNESS IN UNIFORM-SPACES AND FUZZY UNIFORM-SPACES

The notion of a weak-Cauchy ultrafilter in a uniform space is defined and a useful characterisation theorem obtained. It is then shown that a subset is bounded iff every ultrafilter on the set is weak-Cauchy. The properties of bounded sets can then be established using this theorem. Turning to fuzzy uniform spaces, the notion of a weak-Cauchy prime prefilter is defined and the theory of these prefilters is developed. The notion of a bounded fuzzy set is defined and it is shown that it is a good extension of the standard notion. The theory of bounded fuzzy sets is then obtained and it is seen to be a generalisation of the theory of bounded sets in a uniform space.