# Links between Contractibility and Fixed Point Property for Khalimsky Topological Spaces

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (1)
- Distinct points $x,y\in X$ are said to be K-adjacent if $x\in S{O}_{K}\left(y\right)$ or $y\in S{O}_{K}\left(x\right)$.
- (2)
- We say that a sequence ${\left({x}_{i}\right)}_{i\in {[0,l]}_{\mathbb{Z}}},l\ge 2$ 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 $i\in {[0,l]}_{\mathbb{Z}}$. The number l is called the length of this path.
- (3)
- We say that an (injective) sequence ${\left({x}_{i}\right)}_{i\in {[0,l]}_{\mathbb{Z}}}$ in X is a simple K-path if ${x}_{i}$ and ${x}_{j}$ are K-adjacent if and only if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=1$.
- (4)
- A simple closed K-curve with l elements in ${\mathbb{Z}}^{n},n\ge 2,l\ge 4$, denoted by $S{C}_{K}^{n,l}$, is a simple K-path (or just a sequence) ${\left({x}_{i}\right)}_{i\in {[0,l-1]}_{\mathbb{Z}}}$ in ${\mathbb{Z}}^{n}$ such that ${x}_{i}$ and ${x}_{j}$ are K-adjacent if and only if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=\pm 1\left(mod\phantom{\rule{0.166667em}{0ex}}l\right)$.

## 3. K-Homotopies and K-Homeomorphisms

- (1)
- For any set $X\subset {\mathbb{Z}}^{n}$, the set of spaces $(X,{\kappa}_{X}^{n})$ as objects of KTC denoted by $\mathit{Ob}\left(\mathit{KTC}\right)$;
- (2)
- for all pairs of elements in $\mathit{Ob}\left(\mathit{KTC}\right)$, the set of all K-continuous maps between them as morphisms.

**Definition**

**2.**

**Proposition**

**1.**

**Example**

**1.**

**Definition**

**3.**

**Example**

**2.**

**Remark**

**1.**

**Theorem**

**1.**

**Proof.**

## 4. The Non-K-contractibility of $S{C}_{K}^{n,4}$ and the Non-FPP of $S{C}_{K}^{n,l}$

**Definition**

**4.**

**Definition**

**5.**

**Remark**

**2.**

**Definition**

**6.**

**Definition**

**7.**

**Remark**

**3.**

**Theorem**

**2.**

**Proof.**

**Definition**

**8.**

**Theorem**

**3.**

**Proof.**

**Remark**

**4.**

**Remark**

**5.**

**Theorem**

**4.**

**Theorem**

**5.**

**Corollary**

**1.**

**Remark**

**6.**

- (1)
- The set of spaces ${X}_{n,k}$ with digital k-connectivity as objects of KDTC denoted by $\mathit{Ob}\left(\mathit{KDTC}\right)$;
- (2)
- for all pairs of elements in $\mathit{Ob}\left(\mathit{KDTC}\right)$, the set of all $KD$-continuous maps between them as morphisms.

## 5. Homotopies in the Category KAC and a Certain Conjecture Involving the FPP in KAC

**Definition**

**9.**

**Lemma**

**2.**

**Proof.**

**Definition**

**10.**

- (1)
- The set of $KA$-spaces as objects, denoted by $Ob\left(KAC\right)$,
- (2)
- for every ordered pair of objects $(X,{\kappa}_{X}^{{n}_{0}})$ and $(Y,{\kappa}_{Y}^{{n}_{1}})$, the set of all A-maps $f:(X,{\kappa}_{X}^{{n}_{0}})\to (Y,{\kappa}_{Y}^{{n}_{1}})$ as morphisms.

**Theorem**

**6.**

**Proof.**

**Definition**

**11.**

**Remark**

**7.**

**Definition**

**12.**

**Definition**

**13.**

**Definition**

**14.**

**Definition**

**15.**

**Lemma**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Remark**

**8.**

**Corollary**

**2.**

**Proof.**

## 6. Concluding Remark and Further Work

## Funding

## Conflicts of Interest

## References

- Han, S.-E. Contractibility and fixed point property: the case of Khalimsky topological spaces. Fixed Point Theory Appl.
**2016**, 75. [Google Scholar] [CrossRef][Green Version] - Borsuk, K. Theory of Retracts; Polish Scientific Publisher: Warsaw, Poland, 1967. [Google Scholar]
- Lefschetz, S. Topology; Amer. Math. Soc.: New York, NY, USA, 1930. [Google Scholar]
- Lefschetz, S. On the fixed point formula. Ann. Math.
**1937**, 38, 819–822. [Google Scholar] [CrossRef] - Cellina, A. A fixed point theorem for subsets of L
^{1}. Lect. Note Math. (Multifunct. Integands)**1984**, 1091, 129–137. [Google Scholar] - Fryszkowski, A. The generalization of Cellina’s fixed point theorem. Stud. Math.
**1984**, 78, 213–215. [Google Scholar] [CrossRef][Green Version] - Ivashchenko, A.V. Contractible transformations do not change the homology groups of graphs. Discret. Math.
**1994**, 126, 159–170. [Google Scholar] [CrossRef][Green Version] - Ivashchenko, A.V. Representation of smooth surfaces by graphs, Transformations of graphs which do not change the Lefschetz number of graphs. Discret. Math.
**1993**, 122, 219–233. [Google Scholar] [CrossRef][Green Version] - Mariconda, C. Contractibility and fixed point property: The case of decomposable sets. Nonlinear Anal. Theory Methods Appl.
**1992**, 18, 689–695. [Google Scholar] [CrossRef] - Saha, P.K.; Chaudhuri, B.B. A new approach to computing the Lefschetz number. Pattern Recognit.
**1995**, 28, 1955–1963. [Google Scholar] [CrossRef] - Munkres, J.R. Topology; Prentice—Hall Inc.: Upper Saddle River, NJ, USA, 1975. [Google Scholar]
- Kong, T.Y.; Rosenfeld, A. Topological Algorithms for the Digital Image Processing; Elsevier Science: Amsterdam, The Netherlands, 1996. [Google Scholar]
- Han, S.-E. Remarks on the preservation of the almost fixed point property involving several types of digitizations. Mathematics
**2019**, 7, 954. [Google Scholar] [CrossRef][Green Version] - Alexandorff, P. Diskrete Rume. Mat. Sb.
**1937**, 2, 501–518. [Google Scholar] - Khalimsky, E.D. Applications of connected ordered topological spaces in topology. In Proceedings of the Conference of Mathematics, Nice, France, 1–10 September 1970. [Google Scholar]
- Han, S.-E. The k-homotopic thinning and a torus-like digital image in Z
^{n}. J. Math. Imaging Vis.**2008**, 31, 1–16. [Google Scholar] [CrossRef] - Han, S.-E. Continuities and homeomorphisms in computer topology and their applications. J. Korean Math. Soc.
**2008**, 45, 923–952. [Google Scholar] [CrossRef][Green Version] - Han, S.-E. Homotopy equivalence which is suitable for studying Khalimsky n-dimensional spaces. Topol. Appl.
**2012**, 159, 1705–1714. [Google Scholar] [CrossRef][Green Version] - Khalimsky, E. Motion, deformation, and homotopy in finite spaces. In Proceedings of the IEEE International Conferences on Systems, Man, and Cybernetics, Alexandria, VA, USA, 20–23 October 1987; pp. 227–234. [Google Scholar]
- Šlapal, J. Graphs with a path partition for structuring digital spaces. Inf. Sci.
**2013**, 233, 305–312. [Google Scholar] - Han, S.-E. Non-product property of the digital fundamental group. Inf. Sci.
**2005**, 171, 73–91. [Google Scholar] [CrossRef] - Han, S.-E. On the classification of the digital images up to a digital homotopy equivalence. J. Comput. Commun. Res.
**2000**, 10, 194–207. [Google Scholar] - Han, S.-E. Strong k-deformation retract and its applications. J. Korean Math. Soc.
**2007**, 44, 1479–1503. [Google Scholar] [CrossRef][Green Version] - Spanier, E.H. Algebraic Topology; McGraw-Hill Inc.: New York, NY, USA, 1966. [Google Scholar]
- Rosenfeld, A. Continuous functions on digital pictures. Pattern Recognit. Lett.
**1986**, 4, 177–184. [Google Scholar] [CrossRef] - Kiselman, C.O. Digital Geometry and Mathematical Morphology; Lecture Notes; Department of Mathematics, Uppsala University: Uppsala, Sweden, 2002. [Google Scholar]
- Kang, J.M.; Han, S.-E.; Lee, S. The fixed point property of non-retractable topological spaces. Mathematics
**2019**, 7, 879. [Google Scholar] [CrossRef][Green Version] - Han, S.-E. KD-(k
_{0},k_{1})-homotopy equivalence and its applications. J. Korean Math. Soc.**2012**, 47, 1031–1054. [Google Scholar] [CrossRef][Green Version] - Han, S.-E.; Sostak, A. A compression of digital images derived from a Khalimsky topological structure. Comput. Appl. Math.
**2013**, 32, 521–536. [Google Scholar] [CrossRef] - Han, S.-E. Existence of the category DTC
_{2}(k) which is equivalent to the given category KAC_{2}. Ukranian Math. J.**2016**, 76, 1264–1276. [Google Scholar] [CrossRef][Green Version] - Han, S.-E.; Park, B.G. Digital Graph (k0,k1)-homotopy Equivalence and Its Applications. 2003. Available online: http://atlas-conferences.com/c/a/k/b/35.htm (accessed on 1 August 2003).
- Han, S.-E. U(k)- and L(k)-homotopic properties of digitizations of nD Hausdor spaces. Hacet. J. Math. Stat.
**2017**, 46, 124–144. [Google Scholar] - Han, S.-E. Homotopic properties of an MA-digitization of 2D Euclidean spaces. J. Comput. Syst. Sci.
**2018**, 95, 165–175. [Google Scholar] [CrossRef] - Han, S.-E.; Jafari, S.; Kang, J.M. Topologies on ℤ
^{n}which are not homeomorphic to the n-dimensional Khalimsky topological space. Mathematics**2019**, 7, 72. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Comparison between the two K-topological spaces $X:=(X,{\kappa}_{X}^{2})$ and $Y:=(Y,{\kappa}_{Y}^{2})$ in terms of a K-homeomorphism.

**Figure 2.**Several types of K-homotopies in KTC, ${1}_{X}{\simeq}_{K}f$, ${1}_{X}{\simeq}_{K}g$, and $f{\simeq}_{K}h$.

**Figure 3.**The K-contractibility of X need not imply the K-contractibility relative to the singleton $\left\{q\right\}(\subset X),q=(1,1,1)$.

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Han, S.-E. Links between Contractibility and Fixed Point Property for Khalimsky Topological Spaces. *Mathematics* **2020**, *8*, 18.
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Han S-E. Links between Contractibility and Fixed Point Property for Khalimsky Topological Spaces. *Mathematics*. 2020; 8(1):18.
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Han, Sang-Eon. 2020. "Links between Contractibility and Fixed Point Property for Khalimsky Topological Spaces" *Mathematics* 8, no. 1: 18.
https://doi.org/10.3390/math8010018