Properties of space set topological spaces

Abstract

Since a locally finite topological structure plays an important role in the fields of pure and applied topology, the paper studies a special kind of locally finite spaces, so called a space set topology (for brevity, SST) and further, proves that an SST is an Alexandroff space satisfying the separation axiom T0. Unlike a point set topology, since each element of an SST is a space, the present paper names the topology by the space set topology. Besides, for a connected topological space (X,T) with |X| = 2 the axioms T0, semi-T1/2 and T1/2 are proved to be equivalent to each other. Furthermore, the paper shows that an SST can be used for studying both continuous and digital spaces so that it plays a crucial role in both classical and digital topology, combinatorial, discrete and computational geometry. In addition, a connected SST can be a good example showing that the separation axiom semi-T1/2 does not imply T1/2.

Low-level separation axioms from the viewpoint of computational topology

“…For an A(X), we need to point out that the smallest neighborhood of (2.6) is exactly that of Definition 2.10 so that it is the smallest open set of an element of A(X). Furthermore, if |X| ≥ 2, then a connected A(X) cannot be a discrete topological space [8].…”

Hereditary properties of semi-separation axioms and their applications

“…A topological space (X, T) is called an Alexandroff space if every point x ∈ X has the smallest open neighborhood in (X, T) [2]. Motivated by the Alexandroff topological structure [1,2], several kinds of digital topological spaces and locally finite spaces were developed such as an n-dimensional K-topological space [21], an M-topological space, an axiomatic locally finite space [24], a space set topological space [18] and so on [15,21,24,28]. Furthermore, a study of their properties is included in the papers [11, 20-22, 24, 29].…”

“…Since the low-level separation axioms or the semi-separation axioms play important roles in applied topology including digital topology, computational topology and so on, the paper studies their properties on digital topological spaces such as Khalimsky, Marcus-Wyse topological space, axiomatic locally finite space [16,24], space set topology [18], etc.…”

The fixed point property of the smallest open neighborhood of the n-dimensional Khalimsky topological space

“…Besides, we have proved that not every compact K-topological space has the FPP and the AFPP. The recent paper [9] developed a new type of locally finite space motivated by the ALF-space in [19]. However, the notion of a boundary of a given element was missing in Definition 2.5 of [9].…”

“…The recent paper [9] developed a new type of locally finite space motivated by the ALF-space in [19]. However, the notion of a boundary of a given element was missing in Definition 2.5 of [9]. Thus we need to add it as follows: [Definition of a boundary of a given element of an SST(a Space Set Topological space)]: Let C := (X, N, dim) be an AC complex, where X := {c i j |i ∈ M, j ∈ M i }.…”