Abstract
In this paper, generalizations of adherence and convergence of nets and filters on a bi-GTS are introduced and studied. Several properties and interrelations among such adherence and convergence of nets and filters on a bi-GTS are discussed and characterized using graphs of functions. Finally, these results are applied to investigate the behaviour of a generalization of compactness, known as $g_{ij}$-$closedness$ of a bi-GTS.
Highlights
The converse of Theorem 3.3 is true if we take μi as a topology on X
The converse of Theorem 3.5 is true if we take μi as a topology on X
Proof: Let F be a family of μi-closed sets in X with μj-IFIP such that ∩F = ∅
Summary
A net (xα) on a bi-GTS (X, μ1, μ2) with the directed set (Λ, ≥) as a domain is said to (i) (μi, μj)-adhere (i, j = 1, 2 and i = j) at x ∈ X if for each U ∈ μi(x) and each α ∈ Λ, there exists β ∈ Λ such that β ≥ α and xβ ∈ cμj U . Proof: Let x0 be a (μi, μj)-adherent point of a filterbase F and P : Λ → X be the net based on F.
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