Aspects of nearness in σ-frames

Complete nearness σ-frames

“…μ) is separable and strong there exists B ∈ Coz μ such that B h * (A) (see Lemma 3.2 in [11]). Since h is onto,…”

“…Let L be a frame. The following are equivalent for any x ∈ L: and the category of separable, strong Lindelöf nearness frames SepSLNFrm shown in [11]. A separable nearness frame is one in which the countable uniform covers generate the nearness whilst a strong nearness σ-frame is one in which for each uniform cover A there is a uniform cover B such that A B i.e.…”

“…In [18] complete uniform σ-frames were characterised and this notion was modelled for metric σ-frames in [15]. The papers [10] and [11] introduced the structure of a nearness on a σ-frame as a generalization of that of uniformity. % I.…”

On completion in the category SSNσFrm

“…The category SepSLNFrm consists of those nearness frames in which the nearness is separable and strong and where the underlying frame is Lindelöf. The adjunction described above between LRegFrm and RegσFrm extends to the structured nearness setting as described in [12] which we will require in the sequel.…”

“…The structure of a nearness on a σ-frame generalized the notion of uniformity in [10,12] and the category NσFrm of nearness σ-frames and uniform σ-frame homomorphisms was introduced therein. The aim within this paper is to continue with the study on the subcategory SNσFrm of strong nearness σ-frames depicted in [13] where the notion of complete was investigated.…”