Local T2 extended pseudo-quasi-semi metric spaces

Abstract

In this paper, we characterize various local T 2 extended pseudoquasi-semi metric spaces and investigate the relationships among these various forms. Finally, we give some invariance properties of these local T 2 extended pseudo-quasi-semi metric spaces.

Local T_3 Constant Filter Convergence Spaces

“…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 .…”

“…The following are equivalent: Let be a topological category, X is an object of with p∈ ( ). Note that by [3,12] if X is ̅ 0 at p and 2 ′ (resp.…”

“…at p for each p∈ but it is not ̅ 2 at p. Let be a normalized topological category and X be an object of with p∈ ( ). (i) By Theorem 7 of[12], if X is 2 at p, then X is ̅ 2 at p and by Theorems 2.7 and 2.8 of[10], if X is 2 ′ at , then X is ̅ 2 at p. Moreover, by Theorem 2.8 of[10], if X is ̅ 3 (resp. ̅ 3 , 3 ′ , S 3 ′), then X is ̅ 3 at p (resp.…”

T₁ Approach Spaces

“…(1) (i) In category Top of topological spaces and continuous functions as well as in the category SULim semiuniform limit spaces and uniformly continuous maps [24], by Theorem 15 and by Remark 4.7(2) of [8] both T 1 (in our sense) and T 1 (in the usual sense) are equivalent and they reduce to usual T 1 separation axiom. However, in the category pqsMet of extended pseudoquasi-semi metric spaces and non-expensive maps, by Theorem 3.3 of [11], an extended pseudo-quasi-semi metric space (X; d) is T 1 i¤ for all distinct points x; y of X, d(x; y) = 1 and by Theorem 3.4 of [11], (X; d) is T 1 (in the usual sense, i.e., (X; d ) is T 1 , where d is the topology induced from d) i¤ for all distinct points x; y of X, d(x; y) > 0. (ii) By Theorem 11 and Theorem 20, an approach space (X; G) is T 1 if and only if (X; G) is T 1 at p for all p 2 X.…”