Research On Convergent Structure Of Nets And Filters On Preordered Sets

Convergence of sequences is an important topic in analysis and also plays an important role in calculus. The concept of nets is a generalization of that of sequences and a filter is a dual concept of a net. This thesis studies convergence theory of nets and filters in ordered structure. Firstly, it defines and studies upper and lower convergence of nets and filters in preordered sets. It is shown that there is a harmonious relation between upper(resp., lower) convergence of nets and filters and thus they induce a same topology. The related results have some applications in rough set theory, for example, both the open sets and closed sets in the topology induced by the upper convergence or lower convergnce of nets or filters are accurate sets. Secondly, it defines and studies order convergence of nets and filters in preordered sets as well as its relations with upper and lower converngence. Also, there is harmonious relation between net-theoretical order converngence and filter-theoretical order convergence. It is shown that for a net or a filter, it is order convergent to a point if and only if it is both upper convergent and lower convergent to this point. Thirdly, we try an attempt to define and study upper, lower and order convergence of crisp nets and filters in fuzzy preordered set. The results of this paper indicate that there are some combinations among ordered relations, topology and rough sets by means of convergence structures.