THE CATEGORY OF T-CONVERGENCE SPACES AND ITS CARTESIAN-CLOSEDNESS

In this paper, we define a kind of lattice-valued convergence spaces based on the notion of $top$-filters, namely $top$-convergence spaces, and show the category of $top$-convergence spaces is Cartesian-closed. Further, in the lattice valued context of a complete $MV$-algebra, a close relation between the category of$top$-convergence spaces and that of strong $L$-topological spaces is established. In details, we show that the category of strong $L$-topological spaces is concretely isomorphic to that of strong $L$-topological $top$-convergence spaces categorically and bireflectively embedded in that of $top$-convergence spaces.