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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. 1. Motivation and introduction. Lowen [29] pointed out that the category of stratified L-topological spaces is not completely satisfactory from a structural point of view, that is, there is no natural function space structure for the sets of morphisms. In classical theory, this deficiency can be overcome by considering the category of convergence spaces [35], which is a super-category of the category of topological spaces [31]. In the lattice-valued case of the underlying lattice L=[0,1], Lowen et al. [28, 29] consider fuzzy convergence spaces as a generalization of Choquent [2] convergence spaces and the resulting category has, among other things, function space, where prime prefilters play a crucial role as Jäger [22] pointed out. Beside prefilter, there are other kinds of lattice-valued filters in literature of fuzzy topology or nontrivial lattice-valued convergence theory, see Eklund and Gähler [3], Fang [5, 6, 8–10], Flores and others [4], Höhle and Šostak [16], Jäger [21, 22, 24, 25],, Jäger and Burton [23], Li [26, 27], Pang [32–34], Yao [42] and others. Besides Lowen’s function space of fuzzy convergence spaces, there are some significant works of function spaces established by using different lattice-valued filters. For examples: (1) Based on the concept of stratified L-filters [16], Jäger [21, 22] developed a theory of L-generalized convergence spaces in case of L being a complete Heyting algebra, and the resulting category of stratified L-generalized convergence spaces has the function space as desired. And after Jäger’s function space, Fang [5] proposed the category of L-ordered convergence spaces as a reflective full subcategory of the category of L-generalized convergence spaces, which has the function space [11]; (2) By using L-filters of ordinary sets, Yao [42] defined L-fuzzifying convergence spaces and showed the resulting category has the corresponding function space in case of L being a complete Heyting algebra. And after Yao’s work, Wu with coauthor Fang [40] constructed their function spaces in the category of L-ordered fuzzifying convergence spaces, which is considered as a 2010 Mathematics Subject Classification. Primary 54A20; Secondary 54A40

Function spaces based on L-sets