# The Fixed Point Property of Non-Retractable Topological Spaces

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## Abstract

**:**

## 1. Introduction

**[Rival theorem]**For a poset P and let $x\in P$ be irreducible. Then P has the FPP if and only if $P\setminus \left\{x\right\}$ has the FPP.

**[Schröder theorem]**For a poset $(P,\le )$, assume that $a(\in P)$ is retractable to $b\in P$. P has the FPP if and only if (1) $P\setminus \left\{a\right\}$ has the FPP and (2) One of $\{p\in P\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}p>a\}$ and $\{p\in P\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}p<a\}$ has the FPP.

**[Question]**How can we study the FPP of the poset (or the ${T}_{0}$-A-space) in case a given poset (or a ${T}_{0}$-A-space) is related to neither the Rival Theorem nor the Schröder theorem?

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

- (1)
- Two distinct points$x,y\in (X,{\kappa}_{X}^{n}):=X$are called K-path connected (or K-connected) if there is the sequence (or a path)$({x}_{0},{x}_{1},\cdots ,{x}_{l})$on X with$\{{x}_{0}=x,{x}_{1},\cdots ,{x}_{l}=y\}$such that${x}_{i}$and${x}_{i+1}$are K-adjacent,$i\in {[0,l-1]}_{\mathbb{Z}},l\ge 1$. This sequence is called a K-path. Furthermore, the number l is called the length of this K-path.
- (2)
- A simple K-path in X is the K-path${\left({x}_{i}\right)}_{i\in {[0,l]}_{\mathbb{Z}}}$in X such that${x}_{i}$and${x}_{j}$are K-adjacent if and only if$|i-j|=1$.
- (3)
- We say that a simple closed K-curve with l elements${\left({x}_{i}\right)}_{i\in {[0,l-1]}_{\mathbb{Z}}}$in X, denoted by$S{C}_{K}^{n,\phantom{\rule{0.166667em}{0ex}}l},l\ge 4$, is the K-path such that${x}_{i}$and${x}_{j}$are K-adjacent if and only if$|i-j|=\pm 1\left(mod\phantom{\rule{0.166667em}{0ex}}l\right)$.

- The set of objects $(X,{\kappa}_{X}^{n})$, denoted by $Ob\left(KTC\right)$;
- For every ordered pair of objects $(X,{\kappa}_{X}^{{n}_{0}})$ and $(Y,{\kappa}_{Y}^{{n}_{1}})$, the set of all K-continuous maps $f:(X,{\kappa}_{X}^{{n}_{0}})\to (Y,{\kappa}_{Y}^{{n}_{1}})$ as morphisms.

## 3. An Ordered Space Derived from a Khalimsky Topological Space

**Remark**

**1.**

**Proof.**

**Corollary**

**1.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Corollary**

**2.**

## 4. The FPP of Non-$\mathbf{K}$-Retractable Spaces

**Remark**

**2.**

**Proof.**

**Remark**

**3.**

**Definition**

**3**

- (1)
- $(X,{\kappa}_{X}^{n})$is a K-topological subspace of$({X}^{\prime},{\kappa}_{{X}^{\prime}}^{n})$and
- (2)
- $r\left(x\right)=x$for all$x\in X$.

**Example**

**1.**

**Definition**

**4.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

**Example**

**2.**

**Property**

**1.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Example**

**3.**

## 5. Summary and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**a**) (

**1**) $(X,{\kappa}_{X}^{2})$, where $X=\{x,y,z,w,u\}$; (

**2**) The Hasse diagram illustrating the poset derived from $(X,{\kappa}_{X}^{2})$; (

**b**) (

**1**) $(Y,{\kappa}_{Y}^{3})$, where $Y=\{a,b,c,d,e,f,g,h\}$; (

**2**) The Hasse diagram representing the poset derived from $(Y,{\kappa}_{Y}^{3})$ and further, it is not a lattice; (

**c**) Some examples for simple closed K-curves on ${\mathbb{Z}}^{2}$ [28].

**Figure 4.**(

**a**) A non-K-retractable space from the given K-topological plane X; (

**b**) A K-retractable space from the given K-topological plane Y; (

**c**) A K-retractable space from the given K-topological plane Z; (

**d**) A K-retractable space from the given K-topological plane W.

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**MDPI and ACS Style**

Kang, J.M.; Han, S.-E.; Lee, S.
The Fixed Point Property of Non-Retractable Topological Spaces. *Mathematics* **2019**, *7*, 879.
https://doi.org/10.3390/math7100879

**AMA Style**

Kang JM, Han S-E, Lee S.
The Fixed Point Property of Non-Retractable Topological Spaces. *Mathematics*. 2019; 7(10):879.
https://doi.org/10.3390/math7100879

**Chicago/Turabian Style**

Kang, Jeong Min, Sang-Eon Han, and Sik Lee.
2019. "The Fixed Point Property of Non-Retractable Topological Spaces" *Mathematics* 7, no. 10: 879.
https://doi.org/10.3390/math7100879