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Contractibility and fixed point property: the case of Khalimsky topological spaces
Fixed Point Theory and Applications volume 2016, Article number: 75 (2016)
Abstract
Based on the notions of both contractibility and local contractibility, many works were done in fixed point theory. The present paper concerns a relation between digital contractibility and the existence of fixed points of digitally continuous maps. In this paper, establishing a new digital homotopy named by a Khomotopy in the category of Khalimsky topological spaces, we prove that in digital topology, whereas contractibility implies local contractibility, the converse does not hold. Furthermore, we address the following problem, which remains open. Let X be a Khalimsky (K for short) topological space with Kcontractibility. Then we may pose the following question: does the space X have the fixed point property (FPP)? In this paper, we prove that not every Ktopological space with Kcontractibility has the FPP.
Introduction
It is well known that Schauder’s fixed point theorem [1] implies that a nonempty compact convex subset X of a Banach space has a fixed point for any continuous selfmap of X. Before referring to the work, first of all, we need to recall that a topological space X has the FPP if every continuous selfmap f of X has a point \(x \in X\) such that \(f(x)=x\). Since every singleton obviously has the FPP, in studying the FPP of spaces, all spaces X (resp. digital images \((X, k)\)) are assumed to be connected (resp. kconnected) with \(\vert X \vert \geq2\). In relation to the Lefschetz and Borsuk fixed point theorems [2, 3], there was the following conjecture [3]: let X be a contractible and locally contractible space.
Borsuk [2] proved that this conjecture is true in finitedimensional metric spaces. Besides, various cases of the conjecture were proved by Cellina [4] and Fryszkowski [5]. As referred in (1.1), the contractibility of a space X plays an important role in studying the FPP of X and its applications. Thus, many works [2, 4–8] associated with contractibility are well developed.
Digital topology has a focus on studying digital topological properties of nD digital spaces, whereas Euclidean topology deals with topological properties of subspaces of the nD real space, which has contributed to the study of some areas of computer sciences such as computer graphics, image processing, approximation theory, mathematical morphology, optimization theory, and so forth [9–13]. To study digital spaces (see Definition 1), first of all, we have often followed the method established by Rosenfeld [14], the socalled graph theoretical approach (i.e., the Rosenfeld model) [9, 11–15], which is proceeded in many works. Second, one of the wellstudied areas is a Ktopological space [16–18]. A number of properties of the Khalimsky nD space have been also used to study digital spaces [15, 16, 19]. Finally, we have used MarcusWyse (M for short) topology [20–22] to study only 2D digital images.
The present paper develops a Ktopological version of the conjecture (1.1) and some related works posed by Borsuk. At this moment, we need to recall the following differences between metricbased fixed point theory and Ktopologybased fixed point theory. A Ktopological space is not a metric space (see Remark 2.3), contrary to the assumption required by Borsuk. Furthermore, unlike the difference between contractibility and local contractibility in classical mathematics, the present paper proves that their digital versions have their own features (see Theorem 4.6).
In digital topology, there are several types of contractibilities associated with the corresponding digital homotopies [9, 11, 17, 21, 23]. After developing a Khomotopy, we prove that whereas in Ktopology contractibility implies local contractibility, the converse does not hold. Similarly, we prove that whereas kcontractibility of a digital image \((X, k)\) implies local kcontractibility, the converse does not hold.
Rosenfeld (see Theorems 3.3 and 4.1 of [14]) first proved that (for more details, see [24–26])
This means that only a singleton has the FPP in digital topology in a graphtheoretical approach. Nevertheless, Ege and Karaca [27] recently studied the property (1.2) in a graphtheoretical approach (see Theorem 3.8 of [27]). However, the result is proved invalid [25, 26, 28] (see Remark 5.2). Furthermore, to formulate a digital version of the ordinary Lefschetz fixed point theorem in [27], the authors of [27] used digital homology groups of digital images in [27]. However, it turns out that almost of the assertions in [27] are incorrect [24, 26] because the digital version of the Lefschetz number in [27] is not a digital homotopy invariant [26]. Besides, Han [25, 26, 28] recently gave counterexamples to refute this assertion (see Remark 5.2).
Hence, in this paper, we will mainly focus ourselves on studying the FPP of Ktopological spaces instead of digital images \((X, k)\). Besides, we deal only with finite Ktopological spaces (or compact spaces), and we can propose a digital version of (1.1) as a conjecture because contractibility implies local contractibility in digital topology (see Theorem 4.6) as follows: let X be a Ktopological space with Kcontractibility.
To address the conjecture (1.3), the present paper proves that Kcontractibility of a finite Ktopological space need not imply the existence of fixed points of Kcontinuous maps (see Theorems 5.4 and 5.8).
The rest of the paper is organized as follows. Section 2 provides basic notions and terminology from digital topology. Section 3 develops a new digital homotopy named by a Khomotopy to study Kcontractibility. Section 4 investigates various properties of contractibilities in digital topology and compares them. Besides, we develop a digital version of local contractibility and prove that whereas contractibility implies local contractibility, the converse does not hold. Section 5 proves that not every Ktopological space with Kcontractibility has the FPP, which is negative to the conjecture (1.3). But a simple Kpath has the FPP satisfying the property (1.3). Section 6 concludes the paper with summary and further works.
Preliminaries
Let Z, N, and \({\mathbf{Z}}^{n}\) represent the sets of integers, natural numbers, and points in the Euclidean nD space with integer coordinates, respectively. Herman [29] gave the following:
Definition 1
[29]
A digital space is a pair \((X, \pi)\), where X is a nonempty set, and π is a binary symmetric relation on X such that X is πconnected.
In Definition 1, we say that X is πconnected if for any two elements x and y of X, there is a finite sequence \((x_{i})_{i\in[0, l]_{\mathbf{Z}}}\) of elements in X such that \(x=x_{0}\), \(y=x_{l}\), and \((x_{j}, x_{j+1}) \in\pi \) for \(j \in[0, l1]_{\mathbf{Z}}\).
Remark 2.1
In Definition 1, we can consider the relation π according to the situation such as the digital kadjacency relation of (2.1) below and the Kadjacency relation of Definition 2, which are both symmetric relations.
As referred in (1.3), owing to the property (1.2), the present paper mainly studies the FPP from the viewpoint of Ktopology. First, to study the property (1.3), let us recall basic notions and terminology from digital topology such as kadjacency relations of nD integer grids, a digital kneighborhood, digital continuity, and so forth [11–15]. As a generalization of digital kconnectivity of \({\mathbf{Z}}^{n}\), \(n \in\{1, 2 ,3\}\) [12, 13], we will say that two distinct points \(p, q \in{\mathbf{Z}}^{n}\) are kadjacent (or \(k(m,n)\)adjacent) if they satisfy the following property [11] (see also [20, 30]):
For a natural number m, \(1 \leq m \leq n\), two distinct points
are \(k(m, n)\)adjacent (kadjacent for brevity) if
Concretely, these \(k(m, n)\)adjacency relations of \({\mathbf{Z}}^{n}\) are determined according to the number \(m \in{\mathbf{N}}\) [11] (see also [30]).
In terms of the operator (2.1), the kadjacency relations of \({\mathbf{Z}}^{n}\) are obtained [11] (see also [17, 30]) as follows:
where \(C_{i} ^{n}= {n!\over (ni)! i!}\).
For a kadjacency relation of \({\mathbf{Z}}^{n}\), a simple kpath with \(l+1\) elements in \({\mathbf{Z}}^{n}\) is assumed to be an injective sequence \((x_{i})_{i \in[0, l]_{\mathbf {Z}}}\subset{\mathbf{Z}}^{n}\) such that \(x_{i}\) and \(x_{j}\) are kadjacent if and only if \(\vert ij \vert=1\) [12]. If \(x_{0}=x\) and \(x_{l}=y\), then the length of the simple kpath, denoted by \(l_{k}(x, y)\), is the number l. We say that a digital image \((X, k)\) is kconnected if for any two points in X, there is a kpath in X connecting these two points. A simple closed kcurve with l elements in \({\mathbf{Z}}^{n}\), denoted by \(\mathit{SC}_{k}^{n,l}\) [11, 12] (see Figure 1(a)), is the simple kpath \((x_{i})_{i \in[0, l1]_{\mathbf{Z}}}\), where \(x_{i}\) and \(x_{j}\) are kadjacent if and only if \(\vert ij\vert= 1\ ( \operatorname{mod} l)\) [12] (see Figure 1).
Rosenfeld [13] called a set \(X\subset {\mathbf{Z}}^{n}\) with a kadjacency a digital image and denoted it by \((X, k)\). By using the kadjacency relations of \({\mathbf{Z}}^{n}\) of (2.2) we say that a digital kneighborhood of p in \({\mathbf{Z}}^{n}\) is the set [13] \(N_{k}(p):=\{q \mid p \text{ is } k\text{adjacent to } q\}\). Furthermore, we often use the notation [12]
For a digital image \((X, k)\), as a generalization of \(N_{k}^{\ast}(p)\) [12], the digital kneighborhood of \(x_{0} \in X\) with radius ε is defined in X to be the following subset [11] of X:
where \(l_{k}(x_{0}, x)\) is the length of the shortest simple kpath in X from \(x_{0}\) to x, and \(\varepsilon\in\mathbf{N}\). Concretely, for \(X \subset{\mathbf{Z}}^{n}\), we obtain [11]
Second, let us now briefly recall some basic facts and terms related to Ktopology. Motivated by the Alexandroff space [31], the Khalimsky line topology on Z is induced by the set \(\{[2n1, 2n+1]_{\mathbf{Z}}: n \in\mathbf{Z}\}\) as a subbase [31], where for two distinct points a and b in Z, \([a, b]_{{\mathbf{Z}}}= \{n \in{\mathbf{Z}} \mid a\leq n \leq b \}\) [9, 12]. Furthermore, the product topology on \({\mathbf{Z}}^{n}\) induced by \(({\mathbf{Z}}, \kappa)\) is called the Khalimsky product topology on \({\mathbf{Z}}^{n}\) (or Khalimsky nD space), which is denoted by \(({\mathbf {Z}}^{n}, \kappa^{n})\). A point \(x=(x_{1}, x_{2}, \ldots, x_{n}) \in{\mathbf{Z}}^{n}\) is pure open if all coordinates are odd; and it is pure closed if each of the coordinates is even [16]. The other points in \({\mathbf{Z}}^{n}\) are called mixed [16].
For a point \(p:=(p_{1}, p_{2})\) in \(({\mathbf{Z}}^{2}, \kappa^{2})\), its smallest open neighborhood \(\operatorname{SN}_{K}(p)\) is obtained [16]:
where the point \(p:=(p_{1}, p_{2})\) is called closedopen (resp. openclosed) if \(p_{1}\) is even (resp. odd) and \(p_{2}\) is odd (resp. even).
In this paper, each space \(X\subset{\mathbf{Z}}^{n}\) related to Ktopology is considered to be a subspace \((X, \kappa_{X} ^{n})\) induced by \(({\mathbf{Z}}^{n}, \kappa^{n})\) [16, 20].
Let us now recall the structure of \(({\mathbf{Z}}^{n}, \kappa^{n})\). In each of the spaces of Figures 19, a black jumbo dot means a pure open point, and further, the symbols ■ and • mean a pure closed point and a mixed point, respectively. In relation to the further statement of a pure point and a mixed point, we can say that a point x is open if \(\operatorname{SN}_{K}(x) = \{x\}\), where \(\operatorname{SN}_{K}(x)\) means the smallest neighborhood of \(x \in{\mathbf{Z}}^{n}\). Many studies have examined various properties of a Kcontinuous map, connectedness, Kadjacency, a Khomeomorphism [16, 17, 20].
Let us recall the following notions for studying Ktopological spaces.
Definition 2
[20]
Let \((X, \kappa_{X} ^{n}):=X\) be a Ktopological space. We say that two distinct points \(x, y \in X\) are Kadjacent if \(x \in \operatorname{SN}_{K}(y)\) or \(y \in \operatorname{SN}_{K}(x)\). Then we define the following:
We say that a Kpath from x to y in X is a sequence \((x)_{i \in[0, l]_{\mathbf{Z}}}\), \(l \geq2\), in X such that \(x_{0}=x\), \(x_{l}=y\) and each point \(x_{i}\) is Kadjacent to \(x_{i+1}\) and \(i \in[0, l]_{\mathbf{Z}} \). The number l is called the length of this path. A simple Kpath in X is the injective sequence \((x_{i})_{i\in[ 0, l]_{\mathbf{Z}}}\) such that \(x_{i}\) and \(x_{j}\) are Kadjacent if and only if \(\vert ij \vert=1\).
Furthermore, we say that a simple closed Kcurve with l elements in \({\mathbf{Z}}^{n}\), denoted by \(\mathit{SC}_{K}^{n, l}\), \(l\geq4\), is a simple Kpath \((x_{i})_{i \in[0, l1]_{\mathbf{Z}}}\), where \(x_{i}\) and \(x_{j}\) are Kadjacent if and only if \(\vert ij \vert=1\ (\operatorname{mod} l)\).
Example 2.2
In Figure 1(a), \(\mathit{SC}_{4}^{2, 4}\), \(\mathit{SC}_{8}^{2, 4}\), and \(\mathit{SC}_{4}^{2, 8}\) are shown. In Figure 1(b), we have \(\mathit{SC}_{K}^{2, 4}\) and \(\mathit{SC}_{K}^{2, 8}\).
Remark 2.3
Each Ktopological space is not a metric space because it is neither a \(T_{1}\)space nor a regular space although it has a countable basis (see the property (2.5)). Besides, in case we follow a graphtheoretical approach for studying digital spaces (or digital images), a mapping between digital spaces is a graph homomorphism instead of a topological (compact) mapping.
Development of a Khalimsky homotopy and its properties
This section firstly develops the notion of a Khomotopy and investigates various properties of a Khomotopy, which will be used to study both contractibility and local contractibility from the viewpoint of digital topology in Sections 3 and 4. Let us now recall some properties of digital spaces in a graphtheoretical approach. To map every \(k_{0}\)connected subset of \((X, k_{0})\) into a \(k_{1}\)connected subset of \((Y, k_{1})\), the paper [13] established the notion of digital continuity of maps between digital images. Motivated by this approach, the digital continuity of maps between digital images was represented as follows.
Proposition 3.1
Let \((X_{i}, k_{i})\) be digital images in \({\mathbf{Z}}^{n_{i}}\) with the \(k_{i}\)adjacency relations of (2.2), \(i\in\{0, 1\}\). A function \(f: (X_{0}, k_{0}) \to(X_{1}, k_{1})\) is \((k_{0}, k_{1})\)continuous if and only if \(f(N_{k_{0}}(x, 1))\subset N_{k_{1}}(f(x), 1)\) for every \(x\in X_{0}\).
In Proposition 3.1, in case \(k_{1} = k_{2}\), the map f is called a \(k_{1}\)continuous map. By using this concept we establish a digital topological category, denoted by DTC, consisting of two sets [11] (see also [30]):
 •:

for any set \(X \subset{\mathbf{Z}}^{n}\), the set of \((X, k)\) in \({\mathbf{Z}}^{n}\) as objects of DTC;
 •:

for every ordered pair of objects \((X_{i}, k_{i})\), \(i \in\{1, 2\} \), the set of all \((k_{0}, k_{1})\)continuous maps as morphisms of DTC.
Based on the pointed digital homotopy in [9, 16], the following notion of a khomotopy relative to a subset \(A\subset X\) is often used to study a khomotopic thinning and to classify digital images \((X, k)\) in \({\mathbf{Z}}^{n}\) [17, 30].
Definition 3
Let \(((X,A), k_{0})\) and \((Y, k_{1})\) be a digital image pair and a digital image, respectively. Let \(f, g: X \to Y\) be \((k_{0}, k_{1})\)continuous functions. Suppose that there exist \(m \in \mathbf{N}\) and a function \(F:X \times[0, m]_{\mathbf{Z}} \to Y\) such that
 (•1):

for all \(x\in X\), \(F(x, 0)=f(x)\) and \(F(x, m)=g(x)\);
 (•2):

for all \(x \in X\), the induced function \(F_{x} :[0,m]_{\mathbf{Z}} \to Y\) given by \(F_{x}(t)= F(x, t)\) for all \(t \in[0,m]_{\mathbf{Z}}\) is \((2, k_{1})\)continuous;
 (•3):

for all \(t\in[0, m]_{\mathbf{Z}}\), the induced function \(F_{t}: X \to Y\) given by \(F_{t}(x) = F(x,t)\) for all \(x \in X\) is \((k_{0}, k_{1})\)continuous.
 (•4):

Furthermore, for all \(t\in[0, m]_{\mathbf{Z}}\), \(F_{t}(x)=f(x)=g(x)\) for all \(x\in A\).
In Definition 3, if \(A= \{x_{0}\}\subset X\), then we say that F is a pointed \((k_{0}, k_{1})\)homotopy at \(\{x_{0}\}\) [9]. In addition, if \(k_{0} = k_{1}\) and \(n_{0} = n_{1}\), then we say that f and g are pointed \(k_{0}\)homotopic in Y. If, for some \(x_{0}\in X\), \(1_{X}\) is khomotopic to the constant map in the space \(\{x_{0}\}\) relative to \(\{x_{0}\}\), then we say that \((X, x_{0})\) is pointed kcontractible [9, 11].
Remark 3.2
As for the function \(F:X \times[0, m]_{\mathbf{Z}} \to Y\) of Definition 3, the Cartesian product \(X \times[0, m]_{\mathbf{Z}}\) is just a set without any consideration of a digital adjacency for a Cartesian product. In other words, the set \(X \times[0, m]_{\mathbf {Z}}\) is assumed to be a disjoint union \(X \times\{i\}\), \(i\in[0, m]_{\mathbf{Z}}\).
The following notion of a digital homotopy equivalence was firstly introduced in [10, 32] to classify digital images in DTC.
Definition 4
In DTC, for two digital images \((X, k_{0})\) and \((Y, k_{1})\), if there are a \((k_{0}, k_{1})\)continuous map \(h: X \to Y \) and a \((k_{1}, k_{0})\)continuous map \(l:Y \to X \) such that \(l \circ h \) is \(k_{0}\)homotopic to \(1_{X}\) and \(h \circ l \) is \(k_{1}\)homotopic to \(1_{Y}\), then the map \(h: X \to Y\) is called a \((k_{0}, k_{1})\)homotopy equivalence. In this case, we use the notation \(X \simeq_{(k_{0}, k_{1})\cdot h \cdot e}Y\). Furthermore, if \(k_{0} = k_{1}\) and \(n_{0} = n_{1}\), then we call h a \(k_{0}\)homotopy equivalence, and we use the notation \(X \simeq_{k_{0} \cdot h \cdot e} Y\).
We say that a digital image \((X, k)\) is kcontractible if \(X \simeq_{k \cdot h \cdot e} \{x_{0}\}\) for some point \(x_{0} \in X\).
Motivated by both the khomotopy in Definition 3 and the khomotopy equivalence in Definition 4, their Ktopological versions are obtained (see Definitions 6 and 7) in Ktopology. Let us now recall the Kcontinuity of maps between Ktopological spaces. As usual, for two Ktopological spaces \((X, \kappa_{X}^{n_{0}}):=X\) and \((Y, \kappa_{Y}^{n_{1}}):=Y\), a map \(f:X \to Y\) is called continuous at a point \(x\in X\) if for any open set \(O_{f(x)} \subset Y\) containing the point \(f(x)\), there is an open set \(O_{x} \subset X\) containing the point x such that \(f(O_{x}) \subset O_{f(x)}\). Namely, we can represent it as
because each point x in a Ktopological space X always has \(\operatorname{SN}_{K}(x) \subset X\).
By using spaces \((X, \kappa_{X}^{n}):=X\) and Kcontinuous maps, we have a topological category, denoted by KTC, consisting of the following two sets [20]:

(1)
for any set \(X \subset{\mathbf{Z}}^{n}\), the set of spaces \((X, \kappa_{X}^{n})\) as objects of KTC denoted by \(\operatorname{Ob}(\mathit{KTC})\);

(2)
for all pairs of elements in \(\operatorname{Ob}(\mathit{KTC})\), the set of all Kcontinuous maps between them as morphisms.
To study Ktopological spaces in \({\mathbf{Z}}^{n}\), we need to recall a Khomeomorphism as follows:
Definition 5
For two spaces \((X, \kappa_{X}^{n_{0}}):=X \) and \((Y, \kappa_{Y}^{n_{1}}):=Y\), a map \(h : X \to Y\) is called a Khomeomorphism if h is a Kcontinuous bijection and \(h^{1}: Y \to X\) is Kcontinuous.
In \(({\mathbf{Z}}^{n}, T^{n})\), we say that a simple closed Kcurve with l elements in \({\mathbf{Z}}^{n}\) is a path \((x_{i})_{i\in[0, l1]_{\mathbf{Z}}} \subset{\mathbf{Z}}^{n}\), \(l\geq4\), that is Khomeomorphic to a quotient space of a Khalimsky line interval \([a, b]_{\mathbf{Z}}\) in terms of the identification of the only two end points a and b [20], where both of the numbers a and b in \([a, b]_{\mathbf{Z}}\) are even or odd.
Since the Khalimsky nD topological space is a box product of the Khalimsky line space \(({\mathbf{Z}}, \kappa)\), we obviously obtain the following:
Lemma 3.3

(1)
Put \({\mathbf{Z}}^{n} \times\{i\}:={\mathbf {Z}}_{i}^{n}\), \(i \in{\mathbf{Z}}\). Assume \({\mathbf{Z}}_{i}^{n}\) to be the topological space \(({\mathbf{Z}}_{i}^{n}, \kappa_{{\mathbf{Z}}_{i}^{n}}^{n+1})\). Then for any \(i, j \in2{\mathbf{Z}}\) or \(\{2n+1 \mid n \in{\mathbf {Z}}\}\), we see that \(({\mathbf{Z}}_{i}^{n}, \kappa_{{\mathbf{Z}}_{i}^{n}}^{n+1})\) is Khomeomorphic to \(({\mathbf{Z}}_{j}^{n}, \kappa_{{\mathbf {Z}}_{j}^{n}}^{n+1})\).

(2)
\(({\mathbf{Z}}^{n}, \kappa^{n})\) is assumed to be a proper subspace of \(({\mathbf{Z}}^{n+1}, \kappa^{n+1})\) with the relative topology on \({\mathbf{Z}}^{n}\) induced by \(({\mathbf{Z}}^{n+1}, \kappa^{n+1})\), \(n \in{\mathbf{N}}\).
Proof
(1) Consider the map \(h:({\mathbf{Z}}_{i}^{n}, \kappa _{{\mathbf{Z}}_{i}^{n}}^{n+1}) \to({\mathbf{Z}}_{j}^{n}, \kappa_{{\mathbf {Z}}_{j}^{n}}^{n+1})\) given by \(h(x, i)=(x, j)\), where \(x \in({\mathbf{Z}}_{i}^{n}, \kappa_{{\mathbf {Z}}_{i}^{n}}^{n+1})\). Then h is obviously a Khomeomorphism.
(2) Considering \({\mathbf{Z}}^{n}\) to be \({\mathbf{Z}}^{n} \times\{0\} \subset{\mathbf{Z}}^{n+1}\), \(({\mathbf{Z}}^{n}, \kappa^{n})\) is assumed to be a proper subspace of \(({\mathbf{Z}}^{n+1}, \kappa^{n+1})\) with the relative topology on \({\mathbf{Z}}^{n}\) induced by \(({\mathbf{Z}}^{n+1}, \kappa^{n+1})\), \(n \in{\mathbf{N}}\). □
By Lemma 3.3, we obtain the following:
Proposition 3.4

(1)
Any Kinterval \(([a, b]_{\mathbf{Z}}, \kappa _{[a, b]_{\mathbf{Z}}})\) can be embedded into a simple Kpath in \(({\mathbf{Z}}^{n}, \kappa^{n})\).

(2)
\((X, \kappa_{X}^{n})\) is equivalent to the subspace \(X \times\{0\}\) of \(({\mathbf{Z}}^{n+1}, \kappa^{n+1})\) up to Khomeomorphism.

(3)
\(\mathit{SC}_{K}^{n,l}\) is equivalent to the subspace \(\mathit{SC}_{K}^{n,l} \times\{0\} \) of \(({\mathbf{Z}}^{n+1}, \kappa^{n+1})\) up to Khomeomorphism.

(4)
\(\mathit{SC}_{K}^{n_{1}, l}\) is Khomeomorphic to \(\mathit{SC}_{K}^{n_{2}, l}\) even if \(n_{1}\neq n_{2}\).

(5)
Let X and Y be simple Kpaths with the same elements. Then \((X, \kappa_{X})\) need not be Khomeomorphic to \((Y, \kappa_{Y})\).
Proof
(1) It suffices to prove that any Kinterval \(([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf{Z}}})\) is Khomeomorphic to a certain simple Kpath, denoted by \((x_{i})_{i \in[0,l]_{\mathbf {Z}}}\), in \(({\mathbf{Z}}^{n}, \kappa^{n})\) such that \(\vert ba \vert=l\). Indeed, we can take a subspace \((x_{i})_{i \in[0,l]_{\mathbf{Z}}} \subset({\mathbf{Z}}^{n}, \kappa^{n})\) that is Khomeomorphic to \(([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf{Z}}})\) in terms of the mapping of \(f:([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf{Z}}}) \to (x_{i})_{i \in[0,l]_{\mathbf{Z}}} \subset({\mathbf{Z}}^{n}, \kappa ^{n})\) given by
such that for \(i, j \in[0,l]_{\mathbf{Z}}\) (see Figure 2(a)),
\(x_{i}, x_{j} \in(x_{i})_{i \in[0,l]_{\mathbf{Z}}}:=[a, b]_{\mathbf{Z}}\).
(2) By Lemma 3.3 the proof is completed (see Figures 2(b1), 2(b2), and 2(c)). For instance, consider the space \((X, \kappa_{X}^{2})\) in Figure 2(c1). Furthermore, consider the space \((X\times\{0\}:=X_{0}, \kappa_{X_{0}}^{3})\) in Figure 2(c2). Then we see that \((X, \kappa_{X}^{2})\) is Khomeomorphic to \((X_{0}, \kappa _{X_{0}}^{3})\).
(3) By Proposition 3.4(2) the proof is completed.
(4) Owing to the property of \(\mathit{SC}_{K}^{n, l}\), there is an embedding \(i:\mathit{SC}_{K}^{n, l} \to \mathit{SC}_{K}^{n, l} \times\{0\} \subset{\mathbf{Z}}^{n+1}\). More precisely, take any two Kadjacent points \(x, y \in \mathit{SC}_{K}^{n, l}\). If \(\operatorname{SN}_{K}(x) \ni y\), then we see that \(\operatorname{SN}_{K}(y) =\{y\}\) and, further, \(\sharp(\operatorname{SN}_{K}(x))=3\). Since the cardinalities of \(\mathit{SC}_{K}^{n_{1}, l}:=(x_{i})_{i \in[0,l]_{\mathbf {Z}}}\) and \(\mathit{SC}_{K}^{n_{2}, l}:=(y_{i})_{i \in[0,l]_{\mathbf{Z}}}\) are equal to each other, owing to the properties of \(\mathit{SC}_{K}^{n_{i}, l}\), \(i \in\{1,2\}\), we obtain
where the symbol ♯ means the cardinality of a given set. Then we establish a Khomeomorphism between \(\mathit{SC}_{K}^{n_{i}, l}\), \(i \in\{1, 2\}\), as follows: for the points \(x_{i}\), \(x_{j}\), \(y_{i}\), and \(y_{j}\) in (3.1), consider the mapping
where \(x_{j} \in \operatorname{SN}_{K}(x_{i})\) and \(y_{j} \in \operatorname{SN}_{K}(y_{i})\) if and only if \(\vert ij \vert=1\) and \(i, j \in[0, l]_{\mathbf{Z}}\). Then it is obvious that the mapping of (3.2) is a Khomeomorphism.
(5) Consider two simple Kpaths \((X=[0,2]_{\mathbf{Z}}, \kappa_{X})\) and \((Y=[1,3]_{\mathbf{Z}}, \kappa_{Y})\) (see Figure 2(d)). Whereas \((X=[0,2]_{\mathbf{Z}}, \kappa_{X})\) has only one singleton as a smallest open set, \((Y=[1,3]_{\mathbf{Z}}, \kappa_{Y})\) has two singletons as smallest open sets, which cannot be Khomeomorphic to each other. □
To develop the notion of a Khomotopy in KTC (see Definition 6), consider two Ktopological spaces \(X:=(X, \kappa_{X}^{n})\) and a Khalimsky interval (Kinterval for short) \(([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf{Z}}})\). Then, depending on the given space X, we may consider 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).
Let us now establish the notion of a Khomotopy. Furthermore, consider any \((X, \kappa_{X}^{n})\) and \(([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf {Z}}})\), where \([a, b]_{\mathbf{Z}} \in\{[0, m]_{\mathbf{Z}}, [1, m+1]_{\mathbf{Z}}\}\). Then, by Lemma 3.3 and Proposition 3.4(2) we see that \((X, \kappa_{X}^{n})\) is equivalent to \((X\times\{0\}:=X_{0},\kappa_{X_{0}}^{n+1})\) or \((X\times\{1\}:=X_{1},\kappa _{X_{1}}^{n+1})\) up to Khomeomorphism (see Figure 2(c)) or Figure 2(c2)). Thus, we can now establish the notion of a Khomotopy.
Definition 6
In KTC, for two spaces \(X:=(X, \kappa _{X}^{n_{0}})\) and \(Y:=(Y, \kappa_{Y}^{n_{1}})\), let \(f, g: X \to Y\) be Kcontinuous functions. Suppose that there exist a Kinterval \(([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf{Z}}})\) and a function \(F: X \times[a, b]_{\mathbf{Z}} \to Y\) such that
 (∗1):

for all \(x\in X\), \(F(x, a)=f(x)\) and \(F(x, b)=g(x)\);
 (∗2):

for all \(x \in X\), the induced function \(F_{x} :([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf{Z}}}) \to Y\) defined by \(F_{x}(t)= F(x, t)\) for all \(t \in([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf{Z}}})\) is Kcontinuous;
 (∗3):

for all \(t\in [a, b]_{\mathbf{Z}}\), the induced function \(F_{t}: X \to Y\) defined by \(F_{t}(x) = F(x,t)\) for all \(x \in X\) is Kcontinuous.
Then we say that F is a Khomotopy between f and g, and f and g are Khomotopic in Y, denoted \(f \simeq_{K} g\).
In KTC, we say that a Ktopological space X is Kcontractible if the identity map \(1_{X}\) is Khomotopic in X to a constant map with the space consisting of some point \(x_{0} \in X\).
Remark 3.5
(Comparison between a khomotopy in DTC and a Khomotopy in KTC)
(1) Comparing the Khomotopy in Definition 6 with the khomotopy in DTC (see Definition 3), we find some differences between them (see Remark 3.2).
Owing to the Ktopological structure of \(X:=(X, \kappa_{X}^{n_{0}})\), first of all, the set \(X \times[0, m]_{\mathbf{Z}}\) of Definition 3 and that of Definition 6 are different from each other because the latter has the Ktopological structure. Second, depending on the situation of X in Definition 6, we need to take the number m of \(([0, m]_{\mathbf{Z}}, \kappa_{[0, m]_{\mathbf{Z}}})\) even or odd, so that we do the required process under a Khomotopy as in Definition 6.
For instance, let us assume \((X, \kappa_{X}^{n_{0}})\) of Definition 6 to be either \(([0, 3]_{\mathbf{Z}}, \kappa_{[0, 3]_{\mathbf{Z}}})\) or \(([1, 4]_{\mathbf{Z}}, \kappa_{[1, 4]_{\mathbf{Z}}})\). In case \((X, \kappa_{X}^{n_{0}}):=([0, 3]_{\mathbf{Z}}, \kappa_{[0, 3]_{\mathbf{Z}}})\), we see that the space \([0, 3]_{\mathbf{Z}} \times\{0\}:=X_{0}\) (see Figure 2(b1)) as a subspace of \(({\mathbf{Z}}^{2}, \kappa^{2})\) is Khomeomorphic to \(([0, 3]_{\mathbf{Z}}, \kappa_{[0, 3]_{\mathbf {Z}}})\) (see Figure 2(b1)). Besides, we see that \((X_{0}, \kappa _{X_{0}}^{2})\) is Khomeomorphic to \((X_{1}, \kappa_{X_{1}}^{2})\) (see Figure 2(b1)).
In case \((X, \kappa_{X}^{n_{0}}):=([1, 4]_{\mathbf{Z}}, \kappa_{[1, 4]_{\mathbf{Z}}})\), we see that the space \([1, 4]_{\mathbf{Z}} \times\{0\}:=Y_{0}\) (see Figure 2(b2)) as a subspace of \(({\mathbf{Z}}^{2}, \kappa^{2})\) is Khomeomorphic to \(([1, 4]_{\mathbf{Z}}, \kappa_{[1, 4]_{\mathbf {Z}}})\) (see Figure 2(b2)). Besides, we see that \((Y_{0}, \kappa _{Y_{0}}^{2})\) is Khomeomorphic to \((Y_{1}, \kappa_{Y_{1}}^{2})\) (see Figure 2(b2)).
(2) Consider the space \((X, \kappa_{X}^{2})\) in Figure 2(c1). Then, for \(X\times\{i\}:=X_{i}\), \(i \in[0, 2]_{\mathbf{Z}}\), it is clear that each of the subspaces \((X_{i}, \kappa_{X_{i}}^{3})\) is Khomeomorphic to \((X, \kappa_{X}^{2})\) (see Figure 2(c2)).
Furthermore, owing to the current version of a Khomotopy, the Kcontinuity of the map \(F_{x}(t)= F(x, t)\) of the property (∗2) holds.
(3) Consider the space \((X, \kappa_{X}^{2})\) in Figure 3(c), where \(X:=\{ (0, 0), (1, 1), (2, 1), (3, 1)\}\). Then consider the transformation from \((X, \kappa_{X}^{2})\) to \((Y, \kappa _{Y}^{2})\) as shown in Figure 3(c), where \(Y:=\{(1, 2), (2, 3), (3,3), (4, 3)\}\). Whereas the mapping cannot be a Khomotopy that transforms \((X, \kappa_{X}^{2})\) onto \((Y, \kappa_{Y}^{2})\), it can be an 8homotopy without the Ktopological structure.
To classify Ktopological spaces in terms of a certain homotopy equivalence in KTC, we use the following:
Definition 7
In KTC, for two spaces \((X, \kappa _{X}^{n_{0}}):=X\) and \((Y, \kappa_{Y}^{n_{1}}):=Y\), if there are Kcontinuous maps \(h: X \to Y \) and \(l:Y \to X \) such that \(l \circ h \) is Khomotopic to \(1_{X}\) and \(h \circ l \) is Khomotopic to \(1_{Y}\), then the map \(h: X \to Y\) is called a Khomotopy equivalence, denoted \(X \simeq_{K\cdot h \cdot e}Y\).
We say that a digital space \((X, \kappa_{X}^{n})\) is Kcontractible if \(X \simeq_{K \cdot h \cdot e} \{x_{0}\}\) for some point \(x_{0} \in X\). Up to now, we have studied the notions of a Khomotopy and a Khomotopy equivalence and their properties.
Proposition 3.6
The khomotopy equivalence in DTC and the Khomotopy equivalence in KTC have their own features, where the kadjacency relation is taken from (2.2).
Proof
Let us compare among two homotopies in terms of the pictures in Figure 3. We can see some intrinsic processes depending on the corresponding homotopies.
(1) In Figure 3(a), consider the digital image \((X, 4)\). By using the 4homotopy, we see that \((X, 4)\) is 4homotopy equivalent to \(\mathit{SC}_{4}^{2, 8}\).
(2) In Figure 3(b), consider the Ktopological space \((Y, \kappa _{Y}^{2})\). By using the Khomotopy we see that \((Y, \kappa_{Y}^{2})\) is Khomotopy equivalent to \(\mathit{SC}_{K}^{2, 8}\). □
A relation between digital contractibilities and local contractibilities
The notions of contractibility and locally contractibility play an important role in many areas of mathematics [2, 4, 5, 33]. We say that a contractible space is precisely one with the same homotopy type of a singleton [33]. Furthermore, its digital versions have been developed in Definitions 4 and 7 in DTC and KTC, respectively. In relation to the study of the conjecture (1.3), we need the following:
Definition 8
[7]
A topological space X is said to be locally contractible if it satisfies the following equivalent conditions:

(1)
It has a basis of open subsets each of which is a contractible space under the subspace topology.

(2)
For every \(x \in X\) and every open subset V (∋x) of X, there exists an open subset U (∋x) of X such that \(U \subset V\) and U is a contractible space in the subspace topology derived from V.
In classical mathematics, it is well known that contractible spaces are not necessarily locally contractible nor vice versa [7]. For instance, whereas any CWcomplex is locally contractible and any paracompact manifold is locally contractible [7], they need not be contractible,for example, the nD sphere \(S^{n}\), \(n \in {\mathbf{N}}\). Although the comb space [34] is contractible, it cannot be a locally contractible space. Besides, the cone on the Hawaiian earring space [34] is contractible, but it is not locally contractible.
To deal with the conjecture (1.3), we need to establish digital versions of local contractibilities in DTC and KTC. Motivated by the notion of local contractibility in Definition 8, let us establish their digital versions in DTC and KTC.
Definition 9

(1)
In DTC, a digital image \((X, k)\) is said to be locally kcontractible if every point \(x \in X\) has an \(N_{k}(x, 1)\) that is kcontractible.

(2)
In KTC, a Ktopological space \((X, \kappa_{X}^{n})\) is said to be locally Kcontractible if it has a basis of open subsets each of which is a Kcontractible space under the subspace Ktopology.
Let us recall the digital contractibility from the viewpoint of digital topology in a graphtheoretical approach. In [9, 11], the kcontractibility of some simple closed kcurves (see Figure 4) is proved. Namely, it turns out that \(\mathit{SC}_{2n}^{2n, 4}\) is 2ncontractible [25] and, further, \(\mathit{SC}_{3^{n}1}^{n, 4}\) is \((3^{n}1)\)contractible (in case \(n=2\), see [9, 11], and in case \(n\geq3\), see [30]); see Figure 4.
Proposition 4.1
Every digital space in DTC or KTC is locally contractible.
Proof
(1) In DTC, since each point x of a digital image \((X, k)\) has \(N_{k}(x, 1)\) (see (2.4)) which is always kcontractible, the proof is completed.
(2) In KTC, each point x of a Ktopological space \((X, \kappa_{X}^{n})\) has \(\operatorname{SN}_{K}(x)\) (see (2.5)) which is Kcontractible. To be specific, depending on the point \(x \in{\mathbf{Z}}^{n}\), we have its smallest open neighborhood \(\operatorname{SN}_{K}(x)\) (see (2.5) for the case of \(({\mathbf{Z}}^{2}, \kappa^{2})\)) that is Kcontractible (see Figure 5). More precisely, based on Figure 5, consider the maps on \(\operatorname{SN}_{K}(p)\) for the cases of \(({\mathbf{Z}}^{2}, \kappa^{2})\):
Then it is clear to see that the maps F and G are Khomotopies on \(\operatorname{SN}_{K}(x_{0})\) and \(\operatorname{SN}_{K}(y_{0})\), respectively. Furthermore, it is obvious that they make both \(\operatorname{SN}_{K}(x_{0})\) and \(\operatorname{SN}_{K}(y_{0})\) Kcontractible.
By using the method similar to the case of \(({\mathbf{Z}}^{2}, \kappa^{2})\) we can prove the Kcontractibility of \(\operatorname{SN}_{K}(p)\) in \(({\mathbf{Z}}^{n}, \kappa^{n})\). □
Let us investigate some properties of Kcontractibility in KTC.
Lemma 4.2
Any Kpath in \(({\mathbf{Z}}^{n}, \kappa^{n})\) is Kcontractible.
Proof
We will proceed in two steps.
Step 1. Let us consider a Kpath in \({\mathbf{Z}}^{n}\), denoted by \(X:=(x_{i})_{i \in[0, l]_{\mathbf{Z}}}\), as a subspace induced by \(({\mathbf{Z}}^{n}, \kappa^{n})\). Then it is obvious that X contains a simple Kpath \((x_{i}^{\prime})_{i \in[0, l^{\prime}]_{\mathbf{Z}}}:=X^{\prime}\subset X\) with \(l^{\prime}\leq l\). If \(X \setminus X^{\prime}\) is nonempty, then take \(x_{j} \in X \setminus X^{\prime}\) such that \(x_{j} \in \operatorname{SN}_{K}(x_{i})\), where \(x_{i} \in X^{\prime}\), that is, \(x_{i}\) and \(x_{j}\) are Kadjacent to each other. Then consider the map
given by
Then this map F is a Khomotopy (see the process of \(F(x, 1)\) in Figure 7).
Step 2. Since \(X^{\prime}\) is a simple Kpath, by Proposition 3.4 we have a Kinterval \(([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf {Z}}})\) that is Khomeomorphic to \(X^{\prime}:=(x_{i}^{\prime})_{i \in[0, l]_{\mathbf{Z}}}\), where \(([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf{Z}}})\) is Khomeomorphic to the subspace \(([0, l]_{\mathbf{Z}}, \kappa_{[0, l]_{\mathbf{Z}}})\) or \(([1, l+1]_{\mathbf{Z}}, \kappa_{[1, l+1]_{\mathbf {Z}}})\) where the cardinality of \([a, b]_{\mathbf{Z}}\) is equal to that of \([0, l]_{\mathbf{Z}}\) or \([1, l+1]_{\mathbf{Z}}\), that is, \(ba=l\). It is obvious that the Kcontractibility of a simple Kpath is equivalent to the Kcontractibility of \(([0, l]_{\mathbf{Z}}, \kappa _{[0, l]_{\mathbf{Z}}})\) or \(([1, l+1]_{\mathbf{Z}}, \kappa_{[1, l+1]_{\mathbf{Z}}})\). Hence, it suffices to prove that the identity map \(1_{[0, l]_{\mathbf {Z}}}\) on \(([0, l]_{\mathbf{Z}}, \kappa_{[0, l]_{\mathbf{Z}}})\) is Khomotopic to the constant function \(C_{\{0\}}\) given by \(C_{\{0\}}(x)=0\) for all \(x \in[0, l]_{\mathbf{Z}}\) because the proof of the Kcontractibility of \(([1, l+1]_{\mathbf {Z}}, \kappa_{[1, l+1]_{\mathbf{Z}}})\) is similar to that of \(([0, l]_{\mathbf{Z}}, \kappa_{[0, l]_{\mathbf{Z}}})\).
Since the number l is finite, for some \(m\in{\mathbf{N}}\) and any \(s \in[0, l]_{\mathbf{Z}}\), define the map (see Figures 6(a) and 6(b))
given by
It is clear that H is a Khomotopy between \(1_{[0, l]_{\mathbf{Z}}}\) and the constant map \(C_{\{0\}}\), which is the trivial identity map on the singleton \(\{0\}\).
For instance, let us consider the Kintervals \(([0, 3]_{\mathbf{Z}}, \kappa_{[0, 3]_{\mathbf{Z}}})\) and \(([0, 4]_{\mathbf{Z}}, \kappa_{[0, 4]_{\mathbf{Z}}})\) (see Figure 6(a)). Then, in terms of the process from (1) to (4) shown in Figures 6(a) and 6(b), the Kintervals \(([0, 3]_{\mathbf{Z}}, \kappa_{[0, 3]_{\mathbf {Z}}})\) and \(([0, 4]_{\mathbf{Z}}, \kappa_{[0, 4]_{\mathbf{Z}}})\) are proved to be Kcontractible.
Concretely, combining Steps 1 and 2, for some \(m\in{\mathbf{N}}\), we obtain the map
given by (see the process with combined \(F(x, 1)\) and \(H(x, i)\), \(i \in[1, 4]_{\mathbf{Z}}\), in Figure 7)
Then we see that G is a Khomotopy between \(1_{(X, \kappa_{X}^{n})}\) and \(C_{\{x_{0}\}}\), which implies the Kcontractibility of a Kpath. □
Lemma 4.3
\(\mathit{SC}_{K}^{2, 4}\) is Kcontractible.
Proof
The process presented in Figures 8(a) and 8(b) explains the following Kcontractibility of \(\mathit{SC}_{K}^{2,4}\). Motivated by Proposition 3.4(3), let us consider the map (see Figures 8(a) and 8(b)(2))
such that
At this moment, in Figure 8(b)(1), we see that \(\mathit{SC}_{K}^{2,4} \times\{ 0\} \simeq_{K} \mathit{SC}_{K}^{2,4} \times\{1\}\simeq_{K} \mathit{SC}_{K}^{2,4} \times \{2\}\). Then it is obvious that the map F (see (4.3)) is a Khomotopy supporting the Khomotopy equivalence between \(\mathit{SC}_{K}^{2,4}\) and the singleton \(\{c_{0}\}\), which implies that \(\mathit{SC}_{K}^{2,4}\) is Kcontractible.
□
By using the method given by (4.2) we obtain the following:
Corollary 4.4
A Kconnected proper subset of \(\mathit{SC}_{K}^{n,l}\) is Kcontractible.
Proof
By using the method similar to (4.2), we see that a Kconnected proper subset of \(\mathit{SC}_{K}^{n,l}\) is Kcontractible. □
Motivated by nonkcontractibility of \(\mathit{SC}_{k}^{n, l}\), \(l \gneq4\) [11], we obtain the following:
Lemma 4.5
\(\mathit{SC}_{K}^{n, l}\) is not Kcontractible if \(l\gneq4\).
Proof
Let us consider \(\mathit{SC}_{K}^{2, l}\), \(l\gneq4\) (see the spaces W and Z in Figure 9(b) as \(\mathit{SC}_{K}^{2, 8}\)). Then there is at least a part inside of \(\mathit{SC}_{K}^{2, l}\) consisting of two points, a pure point and a mixed point, which are Kadjacent. Due to the part, there is no Khomotopy making \(\mathit{SC}_{K}^{2, l}\) Kcontractible.
By using the method similar to nonKcontractibility of \(\mathit{SC}_{K}^{2, l}\), \(l\gneq4\), we prove the nonKcontractibility of \(\mathit{SC}_{K}^{n, l}\), \(l\gneq 4\). □
Theorem 4.6
The digital contractibility implies the local contractibility. The converse does not hold.
Proof
Owing to Proposition 4.1, since every digital space is locally contractible, it suffices to prove that the local contractibility does not imply contractibility in DTC and KTC.
(1) In DTC, consider \(\mathit{SC}_{k}^{n, l}\) such as \(\mathit{SC}_{8}^{2, 6}\) that is not kcontractible. By Proposition 4.1, whereas it is locally kcontractible, it is not kcontractible.
(2) In KTC, consider \(\mathit{SC}_{K}^{n, l}\) such as \(\mathit{SC}_{K}^{2, 8}\) (see Figure 9(b)) that is not Kcontractible. By Proposition 4.1, whereas it is locally Kcontractible, it is not Kcontractible. □
Contractibility and fixed point property: the case of Khalimsky topological spaces
To study the FPP of digital spaces, we need to recall again that a digital space X (resp. digital image \((X, k)\)) is connected (resp. kconnected) and \(\vert X \vert\geq2\).
Rosenfeld [14] was the first to come up with a fixed point theorem of a digitally continuous selfmap of a digital image \((X, k)\) in \({\mathbf{Z}}^{n}\) with the familiar Euclidean and city block distances. Besides, it was proved in [14] that any digital line segment \(([a, b]_{\mathbf{Z}}, 2)\) does not have the FPP from the viewpoint of digital topology in a graphtheoretical approach, where the cardinality of \([a, b]_{\mathbf{Z}}\) is greater than 1, that is, \(\vert[a, b]_{\mathbf{Z}}\vert\geq2\). This property can be proved as follows. Take two distinct 2adjacent points such as \(x_{i}\) and \(x_{j}\) in \(([a, b]_{\mathbf{Z}}, 2)\). Then, for convenience, we may assume that \(x_{i}\) is even and \(x_{j}\) is odd. Consider the selfmap f of \(([a, b]_{\mathbf{Z}}, 2)\), as follows: for any even numbers \(x \in[a, b]_{\mathbf{Z}}\), \(f(x)=x_{j}\), and the other odd numbers in \([a, b]_{\mathbf{Z}}\) are mapped by the map f into the set \(\{x_{i}\}\). Namely, the image \(f([a, b]_{\mathbf{Z}})\) has the cardinality 2. Then it is clear that the given map f is a 2continuous map that has no fixed points.
For the case of digital image \((X, 2n)\) in \({\mathbf{Z}}^{n}\) with \(\vert X\vert\geq2\), using the method similar to the above approach, let us consider a 2ncontinuous selfmap f of a digital image \((X, 2n)\). Take two distinct points \(x_{i}\) and \(x_{j}\) that are 2nadjacent in X. Let \(f(x)=x_{i}\), \(x \neq x_{i}\), and \(f(x_{i})=x_{j}\) [14]. Then we see that whereas the given map f is a 2ncontinuous map, it cannot have any fixed point. Similarly, Rosenfeld [14] proved that any digital image \((X, k)\) with \(\vert X\vert\geq2\) does not have the FPP either (see Proposition 5.1) as follows: take two kadjacent points \(x,y \in X\) in \({\mathbf{Z}}^{n}\) and consider a selfmap f of \((X, k)\) such that, for all \(x_{1} \in X\) such that \(x_{1}\neq x\),
Then, it is obvious that whereas the given map f is a kcontinuous map, it has no fixed points (for more details, see [24–26]).
Proposition 5.1
[14] (see Theorems 3.3 and 4.1 of [14])
A digital image \((X, k)\) in \({\mathbf{Z}}^{n}\) does not have the FPP if X is kconnected and \(\vert X \vert\geq2\).
Motivated by the Lefschetz fixed point theorem in [3], Ege and Karaca [27] (Theorem 3.8 of [27]) studied a fixed point theorem of a kcontinuous map on a kcontractible digital image in DTC as follows. Let \((X, k) \) be a digital image, and let \(f:(X, k) \to(X, k)\) be any kcontinuous map. If \((X, k) \) is kcontractible, then f has a fixed point. However, by Proposition 5.1 it is clear that this assertion is incorrect [24–26]. Thus, by Proposition 5.1 we conclude the following:
Remark 5.2
The conjecture (1.3) is invalid in DTC.
To make the paper selfcontained and to guarantee Remark 5.2, we have a very simple example: consider a bijective selfmap of \(([0, 1]_{\mathbf{Z}}, 2)\) in DTC such that \(f(0)=1\) and \(f(1)=0\) [25, 26]; whereas \(([0, 1]_{\mathbf{Z}}, 2)\) is 2contractible in terms of the property (4.2), from the viewpoint of DTC and further, the map f is a 2continuous map, which implies that f cannot have any fixed point [25, 26].
Let us now move to the conjecture posed in (1.3).
Question
In KTC, is the conjecture (1.3) valid?
We say that a Ktopological space \((X, \kappa_{X}^{n})\) has the FPP if every Kcontinuous selfmap f of X has a point \(x \in X\) such that \(f(x)=x\).
Let us now study some properties of Ktopological spaces from the viewpoint of fixed point theory.
In KTC, we say that a Ktopological invariant is a property of a Ktopological space that is invariant under Khomeomorphisms.
Proposition 5.3
In KTC, the FPP is a Ktopological invariant.
Proof
Suppose that \((X, \kappa_{X}^{n_{0}})\) has the FPP and there exists a Khomeomorphism \(h:(X, \kappa_{X}^{n_{0}}) \to(Y, \kappa_{Y}^{n_{1}})\). Then we prove that \((Y, \kappa_{Y}^{n_{1}})\) has the FPP. To this end, let g be any Kcontinuous selfmap of \((Y, \kappa_{Y}^{n_{1}})\). Then consider the composition \(h \circ f \circ h^{1}:=g: (Y, \kappa _{Y}^{n_{1}}) \to(Y, \kappa_{Y}^{n_{1}})\), where f is a Kcontinuous selfmap of \((X, \kappa_{X}^{n_{0}})\). Owing to the hypothesis, assume that \(x\in X\) is a fixed point for a Kcontinuous selfmap f of \((X, \kappa_{X}^{n_{0}})\). Since h is a Khomeomorphism, there is a point \(y \in Y\) such that \(h(x)=y\). Let us consider the mapping
Then, from (5.2) we obtain \(h(f(x))=g(y)\). Further, by the hypothesis of the FPP of \((X, \kappa_{X}^{n_{0}})\) and the Khomeomorphism between \((X, \kappa_{X}^{n_{0}})\) and \((Y, \kappa_{Y}^{n_{1}})\), we have
which implies that the point \(h(x)\) is a fixed point of the map g, which implies that \((Y, \kappa_{Y}^{n_{1}})\) has the FPP. □
Theorem 5.4
Let X be a simple Kpath in the nD Khalimsky space. Then it has the FPP.
Proof
In [35], it is proved that any bounded Kinterval \(([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf{Z}}})\) has the FPP. Besides, by Proposition 3.4(1) it is obvious that any simple Kpath in the nD Khalimsky space is Khomeomorphic to a certain Kinterval \(([a, b]_{\mathbf{Z}}, \kappa_{[a, b]_{\mathbf {Z}}})\). By Proposition 5.3 we obtain the assertion. □
Example 5.5
Consider the Kinterval \(([0, 2]_{\mathbf{Z}}, \kappa _{[0, 2]_{\mathbf{Z}}})\) and any Kcontinuous selfmaps of \(([0, 2]_{\mathbf{Z}}, \kappa_{[0, 2]_{\mathbf{Z}}})\). Then there are only seven types of Kcontinuous selfmaps of \(([0, 2]_{\mathbf{Z}}, \kappa_{[0, 2]_{\mathbf{Z}}})\) among nine selfmappings. It is obvious that each of them has at least one fixed point.
Corollary 5.6
\(\mathit{SC}_{K}^{n,l}\) does not have the FPP.
Proof
By the property of \(\mathit{SC}_{K}^{n, l}:=(x_{i})_{i \in[0, l1]_{\mathbf{Z}}}\) we obtain that any two Kadjacent points such as \(x_{i}, x_{i+1\ (\operatorname{mod} l)}\), \(i \in[0, l1]_{\mathbf{Z}}\), have the following property:
In (5.3), in case \(x_{i} \in \operatorname{SN}_{K}(x_{i+1\ (\operatorname{mod} l)})\), it is obvious that the cardinality of \(\operatorname{SN}_{K}(x_{i+1\ (\operatorname{mod} l)})\) is three, and in case \(x_{i+1\ (\operatorname{mod} l)} \in \operatorname{SN}_{K}(x_{i})\), we see that the cardinality of \(\operatorname{SN}_{K}(x_{i})\) is three. Thus, the number l should be even and greater than or equal to 4 because these kinds of alternative arrangement of \(x_{i}\), \(x_{i+1\ (\operatorname{mod} l)}\), \(i \in[0, l1]_{\mathbf{Z}}\), are consecutive. Then consider the selfmap f of \(\mathit{SC}_{K}^{n,l}\) given by \(f(x_{i})=x_{i+2\ (\operatorname{mod} l)}\). Then it is clear that f is a Kcontinuous map without any fixed point. □
Example 5.7
Consider two types of \(\mathit{SC}_{K}^{2, 8}\) in Figures 9(b1) and 9(b2). Take the space \(\mathit{SC}_{K}^{2, 8}:=Z\) in Figure 9(b2). Next, consider the selfmap f of \(\mathit{SC}_{K}^{2, 8}:=Z\) given by \(f(z_{i})=z_{i+2\ (\operatorname{mod} 8)}\). Whereas this map f is obviously a Kcontinuous map, it has no fixed points (see \(\mathit{SC}_{K}^{2, 8}\) in Figures 9(b1) and 9(b2)).
Theorem 5.8
In KTC, the conjecture (1.3) is not valid.
Proof
It suffices to propose a counterexample supporting this assertion. Let us consider \(\mathit{SC}_{K}^{n,4}\), \(n \geq2\), such as \(\mathit{SC}_{K}^{2,4}\) (see Figure 9(a)), Then we see that \(\mathit{SC}_{K}^{n,4}\), \(n \geq2\), is Khomeomorphic to \(\mathit{SC}_{K}^{2,4}\). Then, by Lemma 4.3 it is obvious that \(\mathit{SC}_{K}^{n,4}\) is Kcontractible. Consider the selfmap f of \(\mathit{SC}_{K}^{n,4}\) given by
Whereas the map f is obviously Kcontinuous map, it has no fixed points. □
Summary and further works
Developing the notion of Khomotopy in the category of Khalimsky topological spaces, we have developed the notions of contractibility and local contractibility induced by the Khomotopy. Besides, proving that digital contractibilities imply local contractibilities for a Kcontractible space X, we wondered if the space X has the FPP. In this paper, we proved that not every Ktopological space with Kcontractibility has the FPP. More precisely, for \(\mathit{SC}_{K}^{n,l}\), we proved that \(\mathit{SC}_{K}^{n,l}\) does not have the FPP. For instance, we proved that whereas \(\mathit{SC}_{K}^{n,4}\) is Kcontractible, it cannot have the FPP. However, we proved that a simple Kpath has the FPP. In addition, we proved that in KTC the FPP is a Ktopological invariant.
As a further work, we need to study the FPP of the product of two simple Kpaths. Besides, we need to study the FPP for other digital topological spaces.
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Acknowledgements
The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013R1A1A4A01007577).
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Han, S. Contractibility and fixed point property: the case of Khalimsky topological spaces. Fixed Point Theory Appl 2016, 75 (2016). https://doi.org/10.1186/s1366301605668
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MSC
 55N35
 68U10
Keywords
 Schauder’s fixed point theorem
 fixed point property
 contractibility
 local contractibility
 Khalimsky homotopy
 Khalimsky topology