Extensionality and [formula omitted]-connectedness in the category of ⊤-convergence spaces

Abstract

The extensionality of ⊤-convergence spaces is verified for a complete residuated lattice L with the top element ⊤. And also the E-connectedness of ⊤-convergence spaces for a class E of ⊤-convergence spaces is proposed by generalizing Preuss's connectedness of topological spaces. Then we establish a necessary and sufficient condition that for a class K of ⊤-convergence spaces, there exists a class E of ⊤-convergence spaces such that each space of K is E-connected, where we stress the point that the conclusion benefits from the extensionality of the category of ⊤-convergence spaces. We further present a deep relationship between E-connectedness and T1-separation for ⊤-convergence spaces, that is, the E-connectedness of each subset in a ⊤-convergence space implies that of its closure if and only if E precisely is a class of ⊤-convergence spaces being T1-separated, and as a natural result, the product theorem for E-connected ⊤-convergence spaces is obtained.