and Applied Analysis 3 such that a ∈⇓ d. Thus d ∈⇑ a ⊆ U. Therefore D ∩ U ̸ = 0, hence U ∈ σ(P). This proves τ(P) ⊆ σ(P). (2) Now assume that P is strongly continuous. Then P is continuous. In a strongly continuous lattice, the relation ≪ and ⇐ are the same by the Theorem 2.5 of [5]. Also by [2] in a continuous lattice, every Scott open set A satisfies the condition A = ↑≪A, it follows that ever...

In this paper, the concepts of LωH-sets and LωH-closed spaces are proposed in Lω-spaces by means of (αω)−-remote neighborhood family. The characterizations of LωH-sets and LωH-closed spaces are systematically discussed. Some important properties of LωH-closed spaces, such as the LωH-closed spaces is ω-regular closed hereditary, arbitrarily multiplicative and preserving invariance under almost (...

In this paper we investigate in which cases unions of identi$able classes are also necessarily identi$able. We consider identi$cation in the limit with bounds on mindchanges and anomalies. Though not closed under the set union, these identi$cation types still have features resembling closedness. For each of them we $nd n such that (1) if every union of n − 1 classes out of U1; : : : ; Un is ide...

A category $mathbf{C}$ is called Cartesian closed  provided that it has finite products and for each$mathbf{C}$-object $A$ the functor $(Atimes -): Ara A$ has a right adjoint. It is well known that the category $mathbf{TopFuzz}$  of all topological fuzzes is both complete  and cocomplete, but it is not Cartesian closed. In this paper, we introduce some Cartesian closed subcategories of this cat...