Separation properties at p for the topological category of Cauchy spaces

Abstract

An explicit characterization of each of the separation properties Ti, i = 0, 1, Pre T2, and T2 at a point p is given in the topological category of Cauchy spaces. Moreover, specific relationships that arise among the various Ti, i = 0, 1, Pre T2, and T2 structures at p are examined in this category. Finally, we investigate the relationships between generalized separation properties and separation properties at a point p in this category.

Local T_3 Constant Filter Convergence Spaces

“…(i) By Theorems 3.8 and 3.9, (ii) Note that all objects of a set-based arbitrary topological category may be ̅ 2 at p. For example, it is shown, in [13], that all Cauchy spaces [14] are ̅ 2 at p. Also,…”

“…The following are equivalent: Let be a topological category, X is an object of with p∈ ( ). Note that by [3,12] (ii) Note that all objects of a set-based arbitrary topological category may be ̅ 2 at p. For example, it is shown, in [13], that all Cauchy spaces [14] are ̅ 2 at p. Also, 2 ′ at objects could be only discrete objects [15]. (2) Suppose that ( = ∏ ∈ , ) is ̅ 3 at p. Since each ( , ) is isomorphic to a subspace of (B,K), by Part (1), ∀ ∈ , ( , ) is ̅ 3 at .…”

ST2, delta-T2, ST3, delta-T3, Tychono, compact and lambda-connected objects in the category of proximity spaces

“…Various generalizations of the usual separation properties of topology and for an arbitrary topological category over sets separation properties at a point p are given in [2]. Baran [2] defined separation properties first at a point p, i.e., locally (see [3,5,6,10,13,24,25]), then they are generalized this to point free definitions by using the generic element, [22, p. 39], method of topos theory.…”

Local T2 extended pseudo-quasi-semi metric spaces

“…By Theorem 6, (X; d) is KT 2 at p for all p 2 X but (X; d) is not any of LT 2 at p, T 2 at p, and T 0 2 at p. (C) The relationship between each of T 0 0 at p, T 0 at p, T 1 at p and general T 0 0 , T 0 , T 1 are investigated in [13]. (4) Note, also, that all objects of a set-based arbitrary topological category may be KT 2 at p. For example, by Theorem 3.2 and Theorem 3.4 of [17], it is shown that all Cauchy spaces [15] are T 0 0 at p and P reT 2 at p, and consequently, all Cauchy spaces are KT 2 at p. Now we give some invariance properties of local LT 2 , KT 2 , T 2 , and T 0 2 extended pseudo-quasi-semi metric spaces. Let (X; d) be an extended-pseudo-quasi-semi metric space and F be a none empty subset of X.…”