Local pre-Hausdorff extended pseudo-quasi-semi metric spaces

Abstract

In this paper, we characterize local pre-Hausdor¤ extended pseudoquasi-semi metric spaces and investigate the relationships between them. Finally, we show that local pre-Hausdor¤ extended pseudo-quasi-semi metric spaces are hereditary and productive.

Local T2 extended pseudo-quasi-semi metric spaces

“…(2) Note that the full subcategory KT 2 (pqsMet) is isomorphism-closed. By Theorems 6 and 7 of [14] and Theorem 7, the full subcategory KT 2 (pqsMet) is epire- ‡ective in pqsMet. Finally, if (X; d) is the indiscrete extended pseudo-quasi-semi…”

“…B _ p Bg coincide, then X is called a P reT 0 2 object at p. (4) If the initial lift of the U-sources Proof. The proof of (1) and (2) are given in [14].…”

“…(Y; e) is an isomorphism, then it follows easily that d(x; p) = e(f (x); f (p)) for all x; p 2 X, and consequently, by Theorem 6, (X; d) is a T 2 extended pseudo-quasi-semi metric space at p, where T 2 is one of T 2 , T 0 2 , LT 2 . By Theorems 6 and 7 of [14], the subcategory T 2 (pqsMet) is closed under the formation of products and subspaces. Hence, T 2 (pqsMet) is epire ‡ective in pqsMet.…”

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 .…”

“…′ 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 .…”