Function spaces based on L-sets

Abstract

For a commutative, integral, and divisible quantale L,  a concept of top L-convergence spaces  based on L-sets other than crisp sets  is proposed by using a kind of L-filters, namely limited L-filters defined in the paper. Our main result is the existence of function spaces in the the concrete category of top L-convergence spaces over the slice category Set↓L rather than the category Set of sets, such that  the concrete category of top L-convergence spaces over the slice category Set↓L is Cartesian closed. In order to support the existence of  top L-convergence spaces, some nontrivial examples  of  limited L-filters and top L-convergence spaces are presented also.