Low-level separation axioms from the viewpoint of computational topology

Abstract

The present paper studies certain low-level separation axioms of a topological space, denoted by A(X), induced by a geometric AC-complex X. After proving that whereas A(X) is an Alexandroff space satisfying the semi-T1 2 -separation axiom, we observe that it does neither satisfy the pre T1 2 -separation axiom nor is a Hausdorff space. These are main motivations of the present work. Although not every A(X) is a semi-T1 space, after proceeding with an edge to edge tiling (or a face to face crystallization) of Rn, n ? N, denoted by T(Rn) as an AC complex, we prove that A(T(Rn)) is a semi-T1 space. Furthermore, we prove that A(En), induced by an nD Cartesian AC complex Cn = (En,N,dim), is also a semi-T1 space, n ? N. The paper deals with AC-complexes with the locally finite (LF-, for brevity) property, which can be used in the fields of pure and applied mathematics as well as digital topology, computational topology, and digital geometry.