Links between Contractibility and Fixed Point Property for Khalimsky Topological Spaces

: Given a Khalimsky (for short, K -) topological space X , the present paper examines if there are some relationships between the contractibility of X and the existence of the ﬁxed point property of X . Based on a K -homotopy for K -topological spaces, we ﬁrstly prove that a K -homeomorphism preserves a K -homotopy between two K -continuous maps. Thus, we obtain that a K -homeomorphism preserves K -contractibility. Besides, the present paper proves that every simple closed K -curve in the n -dimensional K -topological space, SC n , l K , n ≥ 2, l ≥ 4, is not K -contractible. This feature plays an important role in ﬁxed point theory for K -topological spaces. In addition, given a K -topological space X , after developing the notion of K -contractibility relative to each singleton { x } ( ⊂ X ) , we ﬁrstly compare it with the concept of K -contractibility of X . Finally, we prove that the K -contractibility does not imply the K -contractibility relative to each singleton { x 0 } ( ⊂ X ) . Furthermore, we deal with certain conjectures involving the (almost) ﬁxed point property in the categories KTC and KAC , where KTC (see Section 3) ( resp. KAC (see Section 5)) denotes the category of K -topological ( resp. KA -) spaces, KA -) spaces are subgraphs of the connectedness graphs of the K -topology on Z n .


Introduction
First of all, we recall that in a category an object X has the fixed point property (FPP, for short) if every self-morphism f of X has a point x ∈ X such that f (x) = x. Since every singleton obviously has the FPP, when studying the FPP of topological spaces, each topological space X (resp. digital image (X, k)) is assumed to be connected (resp. k-connected) and | X | ≥ 2. Thus, each set X involving the FPP considered in this paper is assumed to follow this requirement. As stated in [1], associated with the Borsuk and the Lefschetz fixed point theorems [2][3][4], there was the following conjecture [3]: Let X be a contractible and locally contractible space.
Then it has the FPP for compact mappings. (1) Indeed, as mentioned in [1], Borsuk proved in [2] that this conjecture is true in finite dimensional metric spaces. As referred in (1), the contractibility of a metric space X plays an important role in studying the FPP of a metric space X. Thus, many works [2,[5][6][7][8][9] associated with contractibility are well developed. Meanwhile, it is obvious that a K-topological space is not metrizable [1] because it is not even a regular space. Hence it cannot be metrizable, according to Nagata-Smirnov theorem [10]. Hereafter, since we often use the term "Khalimsky", we will use instead of it 'K-' for short if there is no danger of confusion. Indeed, a paper [1] studied the conjecture (1) from the viewpoint of the category

Preliminaries
Let Z, N and Z n represent the sets of integers, natural numbers and points in the Euclidean n-dimensional space with integer coordinates, respectively. Let us now briefly recall some notions related to K-topology. The Khalimsky line topology on Z, as an Alexandroff space [13], is induced by the set {[2n − 1, 2n + 1] Z : n ∈ Z} as a subbase [13]. Furthermore, the product topology on Z n induced by (Z, κ) is called the Khalimsky product topology on Z n (or Khalimsky n-dimensional space) which is denoted by (Z n , κ n ). A point x = (x 1 , x 2 , · · · , x n ) ∈ Z n is pure open if all coordinates are odd; and it is pure closed if each of the coordinates is even [14]. The other points in Z n are called mixed [14]. For a point p := (p 1 , p 2 ) in (Z 2 , κ 2 ), its smallest open neighborhood SO K (p) is obtained, as follows [14]. For m, n ∈ Z, {(p 1 − 1, p 2 ), p, (p 1 + 1, p 2 )} if p = (2m, 2n + 1), {(p 1 , p 2 − 1), p, (p 1 , p 2 + 1)} if p = (2m + 1, 2n), In this paper each space X(⊂ Z n ) related to K-topology is considered to be a subspace (X, κ n X ) induced by (Z n , κ n ) [14,15]. Let us now recall the structure of (Z n , κ n ). In each of the spaces of Figures 1-6, a black jumbo dot means a pure open point and further, the symbols and • mean a pure closed point and a mixed point, respectively. Many studies have examined various properties of a K-continuous map, connectedness, K-adjacency, a K-homeomorphism [14][15][16][17][18][19].
Let us recall the following terminology for studying K-topological spaces. Definition 1. [15] Let X := (X, κ n X ) be a K-topological space.
(1) Distinct points x, y ∈ X are said to be K-adjacent if x ∈ SO K (y) or y ∈ SO K (x).
(2) We say that a sequence (x i ) i∈[0,l] Z , l ≥ 2 in X is a K-path from x to y if x 0 = x, x l = y and each point x i is K-adjacent to x i+1 and i ∈ [0, l] Z . The number l is called the length of this path. (3) We say that an (injective) sequence (x i ) i∈[0,l] Z in X is a simple K-path if x i and x j are K-adjacent if and only if | i − j | = 1. (4) A simple closed K-curve with l elements in Z n , n ≥ 2, l ≥ 4, denoted by SC n,l K , is a simple K-path (or just a sequence) (x i ) i∈[0,l−1] Z in Z n such that x i and x j are K-adjacent if and only if | i − j | = ±1(mod l).
For instance, we can see SC 3,4 K in Figure 4.

K-Homotopies and K-Homeomorphisms
In this section we examine if a K-homeomorphism preserves a K-homotopy between two K-continuous maps. Let us now recall the notion of K-continuity of a map from f : X → Y, where X := (X, κ n 0 X ) and Y := (Y, κ n 1 Y ), as follows: because each point x in a K-topological space X always has SO K (x) ⊂ X, where SO K (x) (resp. SO K ( f (x))) is the smallest open set of x (resp. f (x)) in X(resp. Y).
Using spaces X := (X, κ n X ) and K-continuous maps, we have a topological category, denoted by KTC, consisting of the following two data [15]: (1) For any set X ⊂ Z n , the set of spaces (X, κ n X ) as objects of KTC denoted by Ob(KTC); (2) for all pairs of elements in Ob(KTC), the set of all K-continuous maps between them as morphisms.
To study K-topological spaces, we need to recall a K-homeomorphism as follows: Definition 2. [14,15] For two spaces X := (X, κ n 0 X ) and Y := (Y, κ n 1 Y ), a map h : X → Y is called a K-homeomorphism if h is a K-continuous bijection, and h −1 : Y → X is K-continuous.
Owing to (4), the Alexandroff topological structure of a K-topological space and the bijection of a K-homeomorphism, we obtain the following: In view of (5), we can represent a K-homeomorphism as follows: A map h : X → Y is a K-homeomorphism if and only if h is a bijection satisfying the property h(SO K (x)) = SO K (h(x)) for any point x ∈ X. Example 1. Consider the two K-topological spaces (X, κ 2 X ) and (Y, κ 2 Y ) in Figure 1. Although they have the same shape with the same cardinality, they are not K-homeomorphic. To be precise, for the points p 1 := (0, 0, 0), p 2 := (2, 0, 0) ∈ X, we obtain where | · | mean the cardinality of the given set. However, the space (Y, κ 2 Y ) does not contain any points whose cardinalities are 9 or 11. Thus, we complete the proof contrary to (5). Let us recall the notion of K-homotopy for K-topological spaces. Consider (X, κ n X ) and Then we say that F is a K-homotopy between f and g, and f and g are K-homotopic in Y. In addition, we use the notation f K g.

Example 2.
Consider certain three K-continuous self-maps f , g, and h of X shown in Figure 2 with their Im( f ), Im(g), and Im(h) in Figure 2. Then we observe 1 X K f , 1 X K g, and f K h (the improvement of Figure 3c of [1]).

Remark 1.
In view of the properties ( * 2) and ( * 3) of Definition 3, for the homotopy F : need not be a subspace of the product space (Z n 0 +1 , κ n 0 +1 ). Namely, we may consider it as just a Cartesian product of two K-topological spaces X and ( Let us now examine if a K-homeomorphism preserves a K-homotopy between two K-continuous maps.

Theorem 1. A K-homeomorphism preserves a K-homotopy.
Proof. Suppose a K-homotopy between two K-continuous maps f and g. Namely, given two spaces X := (X, κ n 0 X ), Y := (Y, κ n 1 Y ), and the two K-continuous functions f , g : X → Y, we consider a K-homotopy F : Besides, further assume two K-homeomorphisms h 1 : X → X and h 2 : Y → Y , where X := (X , κ n 0 X ) and Y := (Y , κ n 1 Y ). Indeed, the dimensions n 0 and n 1 of X and Y need not be equal to those of X and Y, respectively. Then, it is obvious that the two composites are also K-continuous maps from X to Y . To be specific, based on the given K-homotopy and the two K-homeomorphisms h 1 and h 2 , let us now define the new map Then, we obtain the following: Indeed, Theorem 1 will be strongly used in Section 4.
(Step 1) It is obvious that the given space X is K-contractible relative to the singleton {p}, i.e., Step 2) We prove that the given space X is not K-contractible relative to the singleton {q} ⊂ X, where Namely, we may consider the point q to be a point in the second level of the set X of Figure 3. Without loss of generality, we may take any point q in (6) and prove that X is not K-contractible relative to the singleton {q}. For convenience, let us consider q := (1, 1, 1). Then we prove X is not K-contractible relative to the singleton {q}. Using the 'reductio ad absurdum', suppose that there is a K-homotopy, F : Let us now consider the point x := (2, 0, 0). Then we obtain According to the K-homotopy satisfying 1 X K·rel.{q} C {q} , based on (6), we may assume the mappings of the point x by F in the following way: In case we follow the mapping (7), we observe that the mapping does not support the property ( * 2) of Definition 3. In case we take the mapping (8), we find that the mapping does not support the property ( * 2) of Definition 3 either. The other cases are similarly proved by using the above method. Thus, we conclude that for the point q of (6), there is no K-homotopy supporting 1 X K·rel.{q} C {q} .  Based on the properties of contractibility, regarding the conjecture (1), we need to deal with the notion of local K-contractibility. As referred in (1), the notions of contractibility and locally contractibility play important role in many areas of mathematics [2,5,6,23]. In typical homotopy theory, we say that a contractible space is precisely one with the same homotopy type of a singleton [23]. In typical mathematics, it is well known that contractible spaces are not necessarily locally contractible nor vice versa [10] (see [1] for more details). To deal with the conjecture (2), we need to recall the K-topological version of the local contractibility, as follows: Definition 8. [1] In KTC, a K-topological space (X, κ n X ) is said to be locally K-contractible if it has a basis of open subsets each of which is a K-contractible space under the subspace K-topology.

Proposition 2. [1] Every space in KTC is locally contractible.
A paper [1] proved the following: For SC n,4 K , n ≥ 2, we prove the following which can be essentially used in Section 4.
Proof. Suppose SC n,4 K is K-contractible. By Remark 2, since a K-homeomorphism preserves the K-contractibility and SC n,4 K is K-homeomorphic to SC 3,4 K in Figure 4, we may suppose the K-contractibility of SC 3,4 K . Then we must prove that there is a K-homotopy making SC 3,4 K K-contractible, i.e., 1 SC 3,4 K K C {x} for a certain point x ∈ SC 3,4 K . For convenience, put SC 3,4 K := {c 0 , c 1 , c 2 , c 3 } (see Figure 4). Take any singleton as a subset of SC 3,4 K for the examination of the K-contractibility of SC 3,4 K . Without loss of generality, we may take a singleton {c 0 } or {c 1 } because the former is pure closed point and the latter is pure open point. Then we prove that 1 SC 3,4 K cannot be K-homotopic to the constant owing to the property of the K-homotopy, F must map SO K (c 2 ) onto SO K (c 1 ) (see the property ( * 3) of Definition 2). Then the homotopy F does not satisfy the property ( * 2) of Definition 2 because of the non-K-continuity of the mapping from the point c 3 to c 1 , contrary to the given K-homotopy (9). Finally, let us now assume that the point c 2 is mapped by the homotopy F onto the point c 3 . Then, using a method similar to the just above case, this case is also proved to be negative. Thus, we conclude that there is no K-homotopy supporting 1 SC 3,4 SC 3, 4 2 Let us now consider another category, the so-called KDTC, which means the KD-topological category in [16,27]. To do this work, let us just recall two concepts for objects and morphisms for this category. For any set X ⊂ Z n , let X n,k be a space (X, κ n X ) with digital k-connectivity [16]. For two spaces X := X n 1 ,k 1 and Y := Y n 2 ,k 2 , a map f : X → Y is called KD-(k 1 , k 2 )-continuous at a point x ∈ X [16] if f is K-continuous at the point x and further, digitally (k 1 , k 2 )-continuous at a point x. In case f is KD-(k 1 , k 2 )-continuous at every point x ∈ X, we say that f is a KD-(k 1 , k 2 )-continuous map. In other words, a map with the KD-(k 1 , k 2 )-continuity is equivalent to the map satifying both K-continuity and the typical digital (k 1 , k 2 )-continuity in [20]. The category KDTC consists of the following two data.
(1) The set of spaces X n,k with digital k-connectivity as objects of KDTC denoted by Ob(KDTC); (2) for all pairs of elements in Ob(KDTC), the set of all KD-continuous maps between them as morphisms.
A paper [27] established the notion of KD-(k 1 , k 2 )-homotopy in the category KDTC (see Definition 6 of [27]) by replacing Obj(KTC) (resp. Mor(KTC)) with Obj(KDTC) (resp. Mor(KDTC)). Based on this replacement, it also formulated the notion of KD-k-contractibility considered as the KDTC-version of Definitions 3 and 6 using ( * 1)-( * 4) in the present paper. Namely, for X := X n,k ∈ Obj(KDTC), X is called KD-kcontractible relative to a certain singleton {x}(⊂ X) (or KD-kcontractible for short) if 1 X is KD-k-homotopic to a constant map C {x} relative to a certain singleton {x}(⊂ X). Then we use the notation 1 X KD·k·rel.{x} C {x} . Then, the paper asserted that the spaces Y, Z ∈ Obj(KDTC) are KD-8contractible (see Example 4.1 of [27]). However, we need to correct it as follows:

Homotopies in the Category KAC and a Certain Conjecture Involving the FPP in KAC
Unlike the conjecture (2) studied in Section 4, let us now consider the conjecture in the more generalized category, the so-called category KAC which is a topological graph version of KTC. Namely, after establishing KAC-versions of the K-contractibility and the local K-contractibility, we may pose a more generalized version of (2) (see (11) and (13)). This approach can facilitate the study of the FPP and the almost fixed point property (AFPP, for short) of some digital spaces. Indeed, the K-homotopy in KTC in Section 4 is focused on studying the FPP for K-topological spaces. Let us now generalize the conjecture (2) with respect to the FPP and the almost fixed point property (AFPP, for brevity) in KAC (see (11) and (13) for more details). To do this work, objects and morphisms in KAC are certainly assumed, as follows: Considering K-topological spaces (X, κ n X ) with K-adjacency (see Definition 9(1)), we call them KA-spaces which are objects of KAC (see Definition 9). Indeed, a KA-space is a K-topological graph with a K-adjacency inherited from a K-topological space (X, κ n X ) (see Definition 9). Besides, regarding morphisms in KAC, we will use the so-called A-maps (see Definition 10). Definition 9. [28] (1) A KA-space is a set X with K-adjacency derived from a K-topological space (X, κ n X ). Namely, a KA-space X is a K-topological graph inherited from the K-topological space (X, κ n X ) with the adjacency between two distinct points introduced in Definition 1(1).
(2) For a KA-space X := (X, κ n X ) and a point p ∈ X, we define a K-adjacency neighborhood of p to be the set AN X (p) := A X (p) ∪ {p} which is called an A-neighborhood of p, where A X (p) = {x ∈ X | x is K-adjacent to p.}.
As mentioned above, since a KA-space X is totally derived from the K-topological space (X, κ n X ), we often denote a KA-space X with X := (X, κ n X ) or X in short. Hereafter, for convenience, in a KA-space X := (X, κ n X ), we will use AN(p) instead of AN X (p) if there is no danger of ambiguity. In view of (3) and the notion of AN(x), we obtain the following: Lemma 2. Given a KA-space X := (X, κ n X ) and a point x ∈ X, Proof. For a KA-space X := (X, κ n X ), for a point x ∈ X, since the proof is completed.
For a KA-space X and a point x ∈ X, since for x ∈ X we always have AN(x) ⊂ X, we can develop an A-map and an A-isomorphism (see Definitions 10 and 11). Definition 10. [28] Given two KA-spaces X := (X, κ n 0 X ) and Y := (Y, κ n 1 Y ), we say that a function f : X → Y is an A-map at x ∈ X if f (AN(x)) ⊂ AN( f (x)).
Furthermore, we say that a map f : X → Y is an A-map if the map f is an A-map at every point x ∈ X.
In view of Definition 10, we observe that an A-map f : X → Y implies a map preserving connected subsets of X into connected ones [28]. For instance, let us consider the self-map f of an SC n,l K := (x i ) i∈[0,l−1] Z , n ≥ 2, such that f (x i ) = x i+1(mod l) , l ≥ 4. Whereas f is an A-map, it is not a K-continuous map [28].
Using both KA-spaces and A-maps, we establish the so-called KA-category [28], denoted by KAC, consisting of the following data.
As observed in the above self-map f of an SC n,l K , comparing a K-continuous map and an A-map, owing to (10), we obtain the following: Theorem 6. (Theorem 4.5 of [28]) Given a map from X := (X, κ n 0 X ) to Y := (Y, κ n 1 Y ), a K-continuous map implies an A-map. But the converse does not hold.
Proof. Owing to (10) and Definition 10, we complete the proof.
Based on the notion of an A-map, we obtain the following: Definition 11. [28] For two KA-spaces X := (X, κ n 0 X ) and Y := (Y, κ n 1 Y ), a map h : X → Y is called an A-isomorphism if h is a bijective A-map (for brevity, A-bijection) and if h −1 : Y → X is an A-map.
Hereafter, we denote an A-isomorphism between KA-spaces X and Y with X ≈ A Y. In view of Definition 11, we obtain the following: Remark 7. An A-isomorphism h : X → Y implies that for any point x ∈ X, h(AN(x)) = AN(h(x)).
In view of Remark 7, we can represent an A-isomorphism as follows: A map h : X → Y is an A-isomorphism if and only if h is a bijection satisfying the property h(AN(x)) = AN(h(x)) for any point x ∈ X. Definition 12. [28] A simple closed KA-curve with l elements in Z n , n ≥ 2, l ≥ 4, denoted by SC n,l A := (x i ) i∈[0, l−1] Z , is an (injective) sequence (x i ) i∈[0, l−1] Z such that x i and x j are K-adjacent if and only if |i − j| = ±1(mod l).
Let us now study an A-homotopy in KAC [29]. For a space X ∈ Ob(KAC), let B be a subset of X. Then (X, B) is called a KA-space pair. Motivated by many kinds of homotopy equivalences [15,17,21,22,27,30], let us consider the notions of an A-homotopy relative to a subset B ⊂ X [29], A-contractibility [29] and an A-homotopy equivalence [29,31,32]. Definition 13. [29,32] Let (X, B) and Y be a space pair and a space in Ob(KAC), respectively. Let f , g : X → Y be A-maps. Suppose there exist m ∈ N and a function F : X × [0, m] Z → Y such that (•1) for all x ∈ X, F(x, 0) = f (x) and F(x, m) = g(x); (•2) for all x ∈ X, the induced function F x : [0, m] Z → Y given by F x (t) = F(x, t) for all t ∈ [0, m] Z is an A-map; (•3) for all t ∈ [0, m] Z , the induced function F t : X → Y given by F t (x) = F(x, t) for all x ∈ X is an A-map. Then we say that F is an A-homotopy between f and g. (•4) Furthermore, for all t ∈ [0, m] Z , assume that F t (x) = f (x) = g(x) for all x ∈ B.
Then we call F an A-homotopy relative to B between f and g, and we say that f and g are A-homotopic relative to B in Y, f A·rel.B g in symbol.
In Definition 13, if B is a certain singleton of X, then we say that F is a pointed A-homotopy at {x 0 }. If, for some x 0 ∈ X, 1 X is A-homotopic to the constant map C {x 0 } relative to {x 0 }, then we say that (X, x 0 ) is pointed A-contractible (A-contractible if there is no danger of ambiguity) [27]. Let us now recall an A-homotopy equivalence and A-contractibility in KAC. Definition 14. [27] In KAC, for two spaces X and Y, if there are A-maps f : X → Y and g : Y → X such that g • f is A-homotopic to 1 X and f • g is A-homotopic to 1 Y , then the map f : X → Y is called an A-homotopy equivalence. We use the notation X A·h·e Y.

Definition 15.
A KA-space X is said to be locally A-contractible if for every x ∈ X and every AN(x)( x) of X is A-contractible.
Owing to Lemma 1 and Theorem 6, it is obvious that a KA-space X is locally A-contractible.

Lemma 3. Every KA-space is locally A-contractible.
Proof. For a KA-space X, for each point x ∈ X, AN(x)( x) is A-contractible in terms of just an A-homotopy with one step.
Let us propose a certain conjecture in KAC which is the KAC-version of (2) in the KTC. Namely, let X be a KA-space with A-contractibility.
Then it has the FPP for A-mappings. (11) Owing to Lemma 1 and Theorem 6, we obtain the following: (1) An A-homotopy in KAC is a generalization of a K-homotopy in KTC.
(2) A-contractibility is a generalization of the K-contractibility relative to a certain singleton (see Definition 7).
Proof. Since an A-homotopy is define by using the properties (•1)-(•3) of Definition 13, after replacing K-continuous maps in Definition 3 with A-maps, owing to Theorem 6, we prove the assertion.
Owing to Lemma 2, Theorem 6, and Remarks 2 and 3, we obtain the following: (1) An A-isomorphism preserves an A-homotopy between two A-maps.
(3) An A-isomorphism preserves an A-contractibility of a KA-space.

Proof.
(1) Using a method similar to the proof of Theorem 6, we can complete the proof. To be specific, after replacing K-continuous maps (resp. K-homeomorphism and K-homotopy) used in Theorem 6 with A-maps (resp. A-isomorphism and A-homotopy), we only follow the proof of Theorem 6, the proof is completed.
(2) Based on the fact (1), the proof is completed.
(3) Owing to the property (1), the proof is also completed.
In KAC, we say that a KA-space X := (X, κ n X ) has the FPP if every self-A-map f of X has a point x ∈ X such that f (x) = x.

Lemma 4. In KAC, the FPP is invariant up to A-isomorphism.
Proof. Consider a KA-space X := (X, κ n 0 X ) with the FPP. With an A-isomorphism i : X → Y, where Y := (Y, κ n 1 Y ), we prove that Y has the FPP. Let f be any self-A-map of Y. Then consider the composite f := i • g • i −1 : Y → Y, where g is a self-A-map of X. Owing to the hypothesis, assume x ∈ X is a fixed point for a self-A-map g of X. Due to the A-isomorphism i, there is a point y ∈ Y such that i(x) = y. Let us consider the mapping Thus, from (12) we obtain i(g(x)) = f (y) and further, owing to the hypothesis of the FPP of X and the A-isomorphism i, we obtain i(g(x)) = i(x) = y = f (y), which implies that the point i(x) := y is a fixed point of the map f , which implies that Y has the FPP.
Using Lemma 4 and the local A-contractibility of a KA-space, we obtain the following: Theorem 7. The conjecture (11) is negative in KAC.
Proof. We prove that SC n,4 A := {c 0 , c 1 , c 2 , c 3 } is A-contractible. To be specific, owing to Lemma 4, we suffice to prove that SC 3,4 A is both locally A-contractible and A-contractible. It is obvious that SC 3,4 A is locally A-contractible because for any point x ∈ SC 3,4 A , AN(x) is obviously A-contractible, e.g., 1 AN(x) A·rel.{x} C {x} . Let us now prove the A-contractibility of SC 3,4 A , as follows: Consider the map (see Figure 6) Then the map H is an A-homotopy on SC 3,4 A making Finally, using Proposition 5, we observe that SC n,4 A is A-contractible. Next, consider a self-map f of SC n,4 A := {c 0 , c 1 , c 2 , c 3 } defined by f (c i ) = c i+1(mod 4) . Then the map is obviously an A-map which does not support the FPP of SC n,4 A . In KAC, we say that a KA-space X := (X, κ n X ) has the almost (or approximate) fixed point property (AFPP, for short) if every self-A-map f of X has a point x ∈ X such that f (x) = x or f (x) is K-adjacent to x. Regarding the conjecture (11) related to FPP, we now propose the following: Let X be a KA-space with A-contractibility.
Then it has the AFPP for A-mappings.
In view of Theorem 7, we obtain the following: Corollary 2. The conjecture (13) is negative.
Proof. Although SC n,4 A := {c 0 , c 1 , c 2 , c 3 } is A-contractible, n ≥ 2, for the self-A-map f of SC n,4 A defined by f (c i ) = c i+2(mod 4) , we find that this map f does not support the AFPP of SC n,4 A .

Concluding Remark and Further Work
We have proved that a K-homeomorphism preserves a K-homotopy, a K-homotopy equivalence and K-contractibility. Besides, we have firstly proved that SC n,l K is not K-contractible, n ≥ 2, l ≥ 4. In addition, we proved that the K-contractibility of X does not implies the K-contractibility relative to each singleton {x 0 }(⊂ X). Using these properties, we confirmed that in KTC, the conjecture (2) can be positive. Indeed, this feature is very different from that of the k-contractibility of a simple closed k-curve followed from the Rosenfeld's approach [24]. In addition, the conjecture (2) is slightly more generalized version of the conjecture (1.3) of [1] because the K-contractibility involving (1.3) of [1] is equal to the the K-contractibility relative to a certain singleton (see Definition 7 in the present paper). Next, we proved that in KAC the conjectures (11) and (13) are negative.
As a further work, after developing new digital topological structures on Z n or a certain space [33], we can propose a new type of homotopy on the newly-established digital topological spaces. Furthermore, we can examine if the conjecture of (2) is positive or not, and we finally use them in applied sciences such as image processing, homotopic thinning and so on.