Filter spaces

Quasi-Completion of Filter Spaces

“…F ∈ C and G ≥ F imply that G ∈ C, then the pair (X, C) is called a filter space and C is called a pre-Cauchy structure on X. If C and D are two pre-Cauchy structures on X, and C ⊆ D then C is finer than D, written C ≥ D. Associated with each pre-Cauchy structure C on a set X, there is a convergence structure q c , defined as The two filters F and G ∈ F(X) are said to be C − linked [3], if there exist a finite number of filters H 1 , H 2 ,. .…”

“…In this section, a different completion is constructed, which yields such a functor on a subcategory of F IL. The T 2 Wyler completion of a filter space (X, C) that was constructed by Kent and Rath [3] had the property that if (X, C) was a c-filter space (Respectively, Cauchy space), then its completion was also c-filter space (respectively, Cauchy space). However, this is not the case for a quasi-completion.…”

“…Note that if (X, C) is a c-filter space (respectively, Cauchy space), then ((X * 1 , C * 1 ), j) is a c-filter space (respectively, Cauchy space). If we identify each x ∈ X with the equivalence class [ẋ] of all filters which are p c -convergent to x, then the quasiWyler completion coincides with ((X * , C * ), j) in [3]. We will refer to the latter completion as the T 2 Wyler completion of (X, C).…”

“…Since then these spaces have been studied by several topologists (see [3], [4], [5], [6], [7]) in the context of their applications to category theory and algebra . Kent and Rath [3] defined an equivalence relation on a filter space (X, C), which led to the construction of its T 2 Wyler completion. However, soon they realised that unlike the completion of Cauchy spaces, there is no finest completion when there are infinite number of equivalence classes (see Proposition 2.4 [3]).…”

“…Kent and Rath [3] defined an equivalence relation on a filter space (X, C), which led to the construction of its T 2 Wyler completion. However, soon they realised that unlike the completion of Cauchy spaces, there is no finest completion when there are infinite number of equivalence classes (see Proposition 2.4 [3]). Attempts have been made in this paper to construct a certain type of weaker completion, called quasi completion of a filter space which may yield a finest such completion in a subcategory of F IL which has all filter spaces as objects.…”