Image of Contractible Space
Han, SE: Sets of fixed points for digital images. Honam Math. J. 37(4), 595-608 (2015) For a topological space X, all are equivalent: this section first develops the concept of a K-homotopy and examines various properties of a K-homotopy, which are used to examine both contractility and local contractibility from the point of view of numerical topology in sections 3 and 4. Let us now recall some of the properties of numerical spaces in a theoretical approach to graphs. In order to map each subset (k_{0})related to (X, k_{0})) into a subset (k_{1})connected to ((Y, k_{1})), article [13] established the concept of digital continuity of maps between digital images. Motivated by this approach, the digital continuity of the maps between the digital images was represented as follows. Second, let`s briefly recall some basic facts and terms related to the K topology. Motivated by Alexandroff`s space [31], the topology of the Khalimsky line on Z is induced by the set ({[2n-1, 2n+1]_{mathbf{Z}}: n inmathbf{Z}}) as a subbase [31], where for two different points a and b in Z, ([a, b]_{{mathbf{Z}}}= {n in{mathbf{Z}} mid aleq n leq b }) [9, 12]. Additionally, the product topology on ({mathbf{Z}}^{n}) induced by (({mathbf{Z}}, kappa)) is designated as the Khalimsky product topology on ({mathbf{Z}}^{n}) (or Khalimsky nD space), which is denoted by ({mathbf {Z}}^{n}, kappa^{n})). A period (x=(x_{1}, x_{2}, ldots, x_{n}) in{mathbf{Z}}^{n}) is purely open if all coordinates are odd; and it is purely closed when each of the coordinates is straight [16]. The other points of ({mathbf{Z}}^{n}) are mixed [16]. In KTC, a K-topological space X is said to be K-contractible if the identity image (1_{X}) K-homotope is in X to a constant map with the space consisting of a point (x_{0} in X).
If in the definition 3 (A= {x_{0}}Subset X), then F is said to be a sharp homotopy (k_{0}, k_{1})) to ({x_{0}}) [9]. Moreover, if (k_{0} = k_{1}) and (n_{0} = n_{1}), then f and g in Y are selectively (k_{0})-homotope. If for some (x_{0}in X) (1_{X}) k-homotope to the constant image in space ({x_{0}}) is relative to ({x_{0}}), then we say that (X, x_{0})) is k-contractable [9, 11]. A connected K set of (mathit{SC}_{K}^{n,l}) is K-contractable. The terms contractibility and local contractibility play an important role in many areas of mathematics [2, 4, 5, 33]. We say that a contractible space is exactly a space with the same type of homotopy as a singleton [33]. In addition, the digital versions have been developed in definitions 4 and 7 of DTC and KTC respectively. In terms of studying conjecture (1.3), we need the following: To be X a simple path K in Khalimsky`s nD space. Then there is the FPP. To develop the concept of a K-homotopy in KTC (see Definition 6), consider two K-topological spaces (X:=(X, kappa_{X}^{n})) and a Khalimsky interval (short K-interval) (([a, b]_{mathbf{Z}}, kappa_{[a, b]_{mathbf{Z}}})).
Depending on the specified X space, we can then look at the product space (X times[0, m]_{mathbf{Z}}:=X^{prime}, kappa_{X^{prime}}^{n+1})) or (X times[1, m+1]_{mathbf {Z}}:=X^{prime}, kappa_{X^{prime}}^{n+1})), that is, ([a, b]_{mathbf {Z}} in{[0, m]_{mathbf{Z}}, [1, m+1]_{mathbf{Z}}}) (see Lemma 3.3). Be (X_{i}, k_{i})) numeric images in ({mathbf{Z}}^{n_{i}}) with adjacency relationships (k_{i}) of (2.2), (iin{0, 1}). A function (f: (X_{0}, k_{0}) to(X_{1}, k_{1})) is (k_{0}, k_{1}))-continuous if (f(N_{k_{0}}(x, 1))subset N_{k_{1}}(f(x), 1)) for each (xin X_{0}). In mathematics, a topological space X is contractable if the identity mapping on X is nullhomototope, that is, if it is homotopic to a constant figure. [1] [2] Intuitively, a contractable space is a space that can be continuously shrunk to a point in that space. For two spaces ((X, kappa_{X}^{n_{0}}):=X ) and ((Y, kappa_{Y}^{n_{1}}):=Y), an image (h: X to Y) is called K-homeomorphism if h is a K-continuous bijection and (h^{-1}: Y to X) K-continuous. for any set (X subset{mathbf{Z}}^{n}) the set of spaces ((X, kappa_{X}^{n})) as KTC objects, identified by (operatorname{Ob}(mathit{KTC})); Let us remember the following terms for the study of K-topological spaces. To answer the conjecture (1.3), the present work proves that the K-contractibility of a finite K-topological space does not necessarily have to imply the existence of fixed points of K-continuous images (see theorems 5.4 and 5.8). In ({mathbf{Z}}^{n}, T^{n})) we say that a simple closed K curve with elements l in ({mathbf{Z}}^{n}) is a path ((x_{i})_{iin[0, l-1]_{mathbf{Z}}} subset{mathbf{Z}}^{n}), (lgeq4), which is K-homeomorph to a quotient space of a Khalimsky line interval ([a, b]_{mathbf{Z}}) with respect to the identification of the only two endpoints a and b [20], where the two numbers a and b in ([a, b]_{mathbf{Z}}) are even or odd. We say that a numeric image (X, k)) is k-contractable if (X simeq_{k cdot h cdot e} {x_{0}}) for a specific point (x_{0} in X).
To classify K-topological spaces with respect to a particular homotopy equivalence in KTC, we use the following: Next, it is clear that the figures F and G K homotopies are on (Operatorname{SN}_{K}(x_{0})) and (Operatorname{SN}_{K}(y_{0})) respectively. In addition, it is obvious that they make both (nom_opérateur{SN}_{K}(x_{0})) and (nom_opérateur{SN}_{K}(y_{0})) K-contractable. This means that only a singleton has FPP in the numerical topology in a graph theory approach. Nevertheless, Ege and Karaca [27] have recently studied property (1.2) in a theoretical approach to graphs (see sentence 3.8 of [27]). However, the result is invalid [25, 26, 28] (see note 5.2). To formulate a numerical version of Lefschetz`s ordinary fixed-point theorem in [27], the authors of [27] used numerical homology groups of digital images in [27]. However, it turns out that almost the claims of [27] are false [24, 26], since the numerical version of the Lefschetz number in [27] is not a numerical homotopy invariant [26]. In addition, Han [25, 26, 28] has recently provided counter-examples to refute this claim (see note 5.2). It is enough to offer a counter-example to support this assertion. If we look at (mathit{SC}_{K}^{n,4}), (n geq2), as (mathit{SC}_{K}^{2,4}) (see Figure 9(a)), then we see that (mathit{SC}_{K}^{n,4}), (n geq2), is K-homeomorph to (mathit{SC}_{K}^{2,4}). Then it is obvious through lemma 4.3 that (mathit{SC}_{K}^{n,4}) K-contractable. For example, consider the auto-mapping f of (mathit{SC}_{K}^{n,4}) given by considering, for example, the intervals K (([0, 3]_{mathbf{Z}}, kappa_{[0, 3]_{mathbf{Z}}})) and ([0, 4]_{mathbf{Z}}, kappa_{[0, 4]_{mathbf{Z}}})) (see Figure 6(a)).
Then, with regard to the process from (1) to (4) shown in Figures 6(a) and 6(b), the intervals K ([0, 3]_{mathbf{Z}}, kappa_{[0, 3]_{mathbf {Z}}})) and (([0, 4]_{mathbf{Z}}, kappa_{[0, 4]_{mathbf{Z}}})) turned out to be K-contractable. Khalimsky, E: Movement, deformation and homotopy in finite spaces. In: Proceedings IEEE International Conferences on Systems, Man, and Cybernetics, pp. 227-234 (1987) ((X, kappa_{X}^{n})) corresponds to the subspace (X times{0}) of ({mathbf{Z}}^{n+1}, kappa^{n+1})) to K-homeomorphism. Each contractable space is connected to the path and easily connected. Moreover, as all upper homotopy groups disappear, any contractible space for any n ≥ 0 is n-connected. The cone on an X part is always contractable. Therefore, any space can be incorporated into a contractable space (which also shows that subspaces of contractible spaces do not need to be contractible). Now let`s look at some properties of K-topological spaces from the perspective of fixed-point theory. It has a base of open subsets, each of which is a contractable space under the topology of the subspace.
(1) In Figure 3(a), consider the digital image ((X, 4)). Using homotopy 4, we see that ((X, 4)) is a homotopy 4 that corresponds to (mathit{SC}_{4}^{2, 8}). (1) In DTC(mathit{SC}_{k}^{n, l}), think like (mathit{SC}_{8}^{2, 6}), which is not k-contractable. According to Proposition 4.1, it is locally k-contractable, but not k-contractionable. The following notion of numerical homotopy equivalence was first introduced in [10, 32] to classify digital images in DTC. .