Axiomatic Digital Topology

Abstract

The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any connected and non-trivial LF space is isomorphic to an abstract cell complex. The paper demonstrates that in an n-dimensional digital space only those of the (a, b)-adjacencies commonly used in computer imagery have analogs among the LF spaces, in which a and b are different and one of the adjacencies is the "maximal" one, corresponding to 3 n 1 neighbors. Even these (a, b)-adjacencies have important limitations and drawbacks. The most important one is that they are applicable only to binary images. The way of easily using LF spaces in computer imagery on standard orthogonal grids containing only pixels or voxels and no cells of lower dimensions is suggested.

Arrangement of Elements of Locally Finite Topological Spaces Up to an Alf-Homeomorphism

“…The author of [11,12] next formulates axioms related to the notion of a boundary. The definition of the topological boundary or of the frontier is as follows: Definition 4.…”

“…In Definition 4 a neighborhood V of an element e ∈ E is understood as a subset of E containing a subset O ⊂ V containing e which is open in the space S in the classical sense. Then the author of [11,12] introduces the notions of the neighborhood relation, opponents and those of thick and thin frontiers. …”

“…The works [11,12] suggested an axiomatic approach to the theory of LFS. An LFS satisfying these axioms will be called an axiomatic locally finite (for short, ALF) space.…”

“…Due to the several axioms for being an ALF space, we can efficiently study locally finite spaces. All related notions are described in [11].…”

“…The paper is built on the papers [8,11,13,17]. In particular, the paper [17] studied various topological and homotopic properties of finite spaces.…”