Some remarks concerning semi-T1/2 spaces

Abstract

In this paper we prove that each subspace of an Alexandroff T 0-space is semi-T 1 2. In particular, any subspace of the folder X n , where n is a positive integer and X is either the Khalimsky line (Z, τ K), the Marcus-Wyse plane (Z 2 , τ MW) or any partially ordered set with the upper topology is semi-T 1 2. Then we study the basic properties of spaces possessing the axiom semi-T 1 2 such as finite productiveness and monotonicity.

STUDY ON TOPOLOGICAL SPACES WITH THE SEMI-T½SEPARATION AXIOM

“…The present paper explains that for every mixed point p in the Khalimsky nD space the singleton {p} is semi-closed so that the Khalimsky nD space satisfies the semi-T 1 2 -separation axiom [4]. Indeed, the recent paper [4] studies some properties of a semi-T 1 2 space and its product property. It turns out that the separation axiom semi-T 1 2 can play an important role in applied topology such as digital topology and domain theory.…”

“…Motivated from these notions, the recent paper [4] points out that the notion of semi-T 1 2 -separation axiom (see Definition 2) is very related to both the Alexandroff topological structure and the T 0 -separation axiom. The present paper explains that for every mixed point p in the Khalimsky nD space the singleton {p} is semi-closed so that the Khalimsky nD space satisfies the semi-T 1 2 -separation axiom [4]. Indeed, the recent paper [4] studies some properties of a semi-T 1 2 space and its product property.…”

“…It turns out that the separation axiom semi-T 1 2 can play an important role in applied topology such as digital topology and domain theory. For instance, the paper [4] proves that Alexandroff T 0 -spaces are semi-T 1 2 spaces and further, the product and the hereditary property of a semi-T 1 2 space was also studied. Even though the paper [4] referred to the semi-T 1 2 structure of the Khalimsky nD space, the present paper investigates semi-open and semi-closed structures of the Khalimsky nD space.…”

“…The present paper consists of two parts. Since the recent paper [4] proved that an Alexandroff T0-space is a semi-T 1 2 -space, the first part studies semi-open and semi-closed structures of the Khalimsky nD space. The second one focuses on the study of a relation between the LS-property of (SC…”

Some Properties of Lattice-Based K- And M-Maps

“…Let us now recall the nD Ktopological space, denoted by (Z n , κ n ). It turns out that this topological space is an Alexandroff space [1] and a semi-T 1 2 space [2]. For a set X ⊂ Z n we consider the subspace (X, κ n X ) induced by (Z n , κ n ).…”