⊤-diagonal conditions and Continuous extension theorem

Abstract

In this paper, we obtain a generalization of Kowalsky diagonal condition and that of Fischer diagonal condition respectively, namely Kowalsky ⊤-diagonal condition and Fischer ⊤-diagonal condition. We show that our Fischer ⊤-diagonal condition assures a complete-MV-algebra-valued convergence space, proposed in this paper, is strong L-topological, and Kowalsky ⊤-diagonal condition assures a principle (or pretopological) complete-MV-algebra-valued convergence space is strong L-topological also. As applications, we give a “dual form” of our Fischer ⊤-diagonal condition and obtain a concept of regular ⊤-convergence space. In addition, we present an extension theorem for continuous maps from a dense subspace to a regular ⊤-convergence space to show that our ⊤-diagonal conditions works indeed.