Equivalent (k0,k1)-covering and generalized digital lifting

Category Which Is Suitable for Studying Khalimsky Topological Spaces With Digital Connectivity

“…Figure 2, i.e. three types of SC 3,6 18 in (a), (b) and (c) of Figure 2, and the other SC 3,6, * 18 . Then, we obtain the following: (1) SC 3,6, * 18 is neither KD-18-, K-18-nor 18-homeomorphic to SC 3,6 18 .…”

“…Proof: (1) While the point a 0 ∈ SC 3,6, * 18 has the smallest open set N * 18 (a 0 , 2), each point x ∈ SC 3,6 18 in (a), (b) and (c) of Figure 2 has the smallest open sets in N * 18 (x, 1) ⊂ SC 3,6 18 in (a), (b) and (c) of Figure 2, respectively. This implies that SC 3,6, * 18 cannot be KD-(18, 18)-, K-18-and 18-homeomorphic to each of the spaces SC 3,6 18 in (a), (b) and (c) of Figure 2 because there cannot be KD-18-, K-18-and 18-continuous maps from SC 3,6, * 18 onto each of the spaces SC 3,6 18 in (a), (b) and (c) of Figure 2.…”

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

“…The notion of incident elements seems perhaps to be similar to the adjacency introduced in [14] (for more details on multi-dimensional case, see [4,6,7]). There is, however, an important difference between them because we do not suppose that all elements have the same number of incident elements.…”

“…Besides, the papers [5,6,8] also developed several kinds of homeomorphisms and isomorphisms for studying digital spaces. Let us now recall the notion of continuous map in the category of ALF spaces, as follows:…”

“…In relation to the study of discrete objects in Z n , we have used many tools from combinatorial topology, graph theory, Khalimsky topology and so forth [4,5,6,10,11]. Motivated by Alexandroff spaces in [1], the Khalimsky nD space, denoted by (Z n , T n ), was established and its study includes the papers [5,9,10,16].…”