Syntopogenous structure on completely distributive lattice and its connectedness

Abstract

In this paper, the general theory of syntopogenous structures on completely distributive lattices is established. The unified question of cotopology, quasi-uniformity and T-structure is investigated. The results of this paper complete the framework of the topological structure on completely distributive lattices and generalize the corresponding theory in general and fuzzy topology. Finally, we examine the connectedness. The following main results about connectedness are obtained: 1. (1) If F:( L 1, S 1)→( L 2, S 2) is an ( S 1, S 2)-continuous GOH (function), and DϵL 1 is S 1-connected element, then F( D) is S 2-connected element. 2. (2) Let Cϵ( L, S) be an S-connected element and C ⩽ D ⩽ C , then D is an S-connected element. 3. (3) ⊗ I ( L i , S i ) is connected iff for any iϵI, ( L i , S i ) is connected.