Almost fixed point property for digital spaces associated with Marcus-Wyse topological spaces

Abstract

The present paper studies almost fixed point property for digital spaces whose structures are induced by Marcus-Wyse (M-, for brevity) topology. In this paper we mainly deal with spaces X which are connected M-topological spaces with M-adjacency (MA-spaces or M-topological graphs for short) whose cardinalities are greater than 1. Let MAC be a category whose objects, denoted by Ob(MAC), are MA-spaces and morphisms are MA-maps between MA-spaces (for more details, see Section 3), and MTC a category of M-topological spaces as Ob(MT C) and M-continuous maps as morphisms of MTC (for more details, see Section 3). We prove that whereas any MA-space does not have the fixed point property (FPP for short) for any MA-maps, a bounded simple MA-path has the almost fixed point property (AFPP for short). Finally, we refer the topological invariant of the FPP for M-topological spaces from the viewpoint of MT C. c 2017 all rights reserved.

Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations

“…Using this property, unlike the shape of a diamond in Lemma 1 and Corollary 1, as a generalization of the non-FPP of a compact M-topological plane [7], we now prove the non-AFPP of a compact M-topological plane, as follows: Theorem 2. A compact M-topological plane does not have the AFPP.…”

“…Remark 1. It is obvious that SC n,l K [4], SC 2,l M [7] and SC n,l k [3] do not have the AFPP in the categories KTC, MTC and DTC, respectively. For instance, for SC n,l K := (x i ) i∈[0,l−1] Z , consider a self-map of SC n,l K such that f (x i ) = x i+2(mod l) .…”

“…Whereas f is a K-continuous map, there is no point x ∈ SC n,l K such that f (x) = x or f (x) is K-adjacent to x [5]. By using a method similar to this approach for SC n,l K , it is obvious that SC 2,l M and SC n,l k do not have the AFPP in DTC and MTC, respectively (see also [7]).…”

The Fixed Point Property of the Infinite M-Sphere

“…Indeed, it is obvious that the usual topological space (R n , U) is not an Alexandroff space. As an Alexandroff topological space [4,5], the M-topological space was proposed [6] and the study of various properties of it includes the papers [1,[6][7][8][9][10][11][12][13]. Regarding digital spaces [14] in Z 2 , we will follow the concept of a digital k-neighborhood of a point p ∈ Z 2 .…”

“…(2) The picture in Figure 2 of [8] is also misprinted with respect to the dotted arrows. The authors correct it with Figure 2(4) instead of Figure 2(3) (check only the dotted arrows).…”