# The Most Refined Axiom for a Digital Covering Space and Its Utilities

## Abstract

**:**

_{0},k

_{1})-isomorphism; unique lifting property; homotopy lifting theorem; digital covering; digital topological imbedding; generalized digital wedge

## 1. Introduction

## 2. Preliminaries

**Proposition**

**1.**

**Definition**

**1.**

## 3. Development of a Digital Topological Imbedding

**Definition**

**2.**

**Remark**

**1.**

- (1)
- The dimension n need not be equal to ${n}^{\prime}$.
- (2)
- The k-adjacency need not be equal to ${k}^{\prime}$-adjacency.
- (3)
- Unlike the typical notion of a topological imbedding [33], the phrase “with respect to the $(k,{k}^{\prime})$-isomorphism h” is strongly required.

**Definition**

**3.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Example**

**1.**

**Corollary**

**2.**

## 4. Characterizations of Several Types of Local $({\mathit{k}}_{\mathbf{0}},{\mathit{k}}_{\mathbf{1}})$-Isomorphisms and Their Relationships with Both the $({\mathit{k}}_{\mathbf{0}},{\mathit{k}}_{\mathbf{1}})$-Continuity and a Surjection

**Definition**

**4.**

**Example**

**2.**

**Definition**

**5.**

**Lemma**

**1.**

**Proof.**

**Definition**

**6**

**.**For two digital images $(X,{k}_{0})$ in ${\mathbb{Z}}^{{n}_{0}}$ and $(Y,{k}_{1})$ in ${\mathbb{Z}}^{{n}_{1}}$, consider a map $h:(X,{k}_{0})\to (Y,{k}_{1})$. Then the map h is said to be a local $({k}_{0},{k}_{1})$-isomorphism if for every $x\in X$, h maps ${N}_{{k}_{0}}(x,1)$ $({k}_{0},{k}_{1})$-isomorphically onto ${N}_{{k}_{1}}(h(x),1)$. If ${n}_{0}={n}_{1}$ and ${k}_{0}={k}_{1}$, then the map h is called a local ${k}_{0}$-isomorphism.

**Remark**

**2.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Example**

**3.**

- (1)
- In Figure 4a, the map $f:(A,26)\to (B,18)$ defined by $f({a}_{i})={b}_{i},i\in {[0,2]}_{\mathbb{Z}}$, is a local $(26,18)$-isomorphism. However, if we replace $(A,26)$ above by the case $(A,18)$ with the map f above, the map f is not a local 18-isomorphism.
- (2)
- In Figure 4b, the map $h:(\mathbb{Z},2)\to (X,26)$ given by $h(t)={x}_{t(mod\phantom{\rule{0.166667em}{0ex}}3)},t\in \mathbb{Z}$, is not a local $(2,26)$-isomorphism. By contrary, suppose that the map h is a local $(2,26)$-isomorphism. Then, take the digital 2-neighborhood of the point 0 with radius 1, ${N}_{2}(0,1)=\{-1,0,1\}$. Then, we should have the $(2,26)$-isomorphism$${h|}_{{N}_{2}(0,1)}:{N}_{2}(0,1)\to {N}_{26}(h(0),1)={N}_{26}({x}_{0},1)=X.\phantom{\rule{2.em}{0ex}}$$However, the map in (9) is not a $(2,26)$-isomorphism because the inverse map ${(h{|}_{{N}_{2}(0,1)})}^{-1}$ is not $(26,2)$-continuous. More precisely, while every point $t\in {N}_{26}({x}_{0},1)$ has the property ${N}_{26}(t,1)={N}_{26}({x}_{0},1)=X$, some two points in ${(h{|}_{{N}_{2}(0,1)})}^{-1}({N}_{26}({x}_{i},1))$ are not 2-adjacent, ${x}_{i}\in X$, which invokes a contradiction to the $(2,26)$-isomorphism in (9). More precisely, see the points “$-1,1$” which are not 2-adjacent, while their corresponding points $h(-1)={x}_{2},h(1)={x}_{1}$ which are 26-adjacent.
- (3)
- In Figure 4c, the map $p:(X,8)\to S{C}_{8}^{2,7}:={({y}_{i})}_{i\in {[0,6]}_{\mathbb{Z}}}$ defined by $p({x}_{i})={y}_{i},i\in {[0,6]}_{\mathbb{Z}}$, is not a local 8-isomorphism, where $X:=\left\{{x}_{i}\phantom{\rule{0.166667em}{0ex}}\right|\phantom{\rule{0.166667em}{0ex}}i\in {[0,6]}_{\mathbb{Z}}\}\subset {\mathbb{Z}}^{2}$ and ${x}_{0}=(0,0),{x}_{1}=(1,1),{x}_{2}=(2,0),{x}_{3}=(3,1),{x}_{4}=(4,0),{x}_{5}=(5,1),{x}_{6}=(4,0)$. Indeed, we may call the set $(X,8)$ a finite fence set with 8-adjacency.

## 5. The Most Refined Axiom for a Digital Covering Space

**Definition**

**7.**

- (1)
- for some index set M, ${p}^{-1}({N}_{{k}_{1}}(b,\epsilon ))={\cup}_{i\in M}{N}_{{k}_{0}}({e}_{i},\epsilon )$ with ${e}_{i}\in {p}^{-1}(b)$;
- (2)
- if $i,j\in M$ and $i\ne j$, then ${N}_{{k}_{0}}({e}_{i},\epsilon )\cap {N}_{{k}_{0}}({e}_{j},\epsilon )$ is an empty set; and
- (3)
- the restriction of p to ${N}_{{k}_{0}}({e}_{i},\epsilon )$ from ${N}_{k}({e}_{i},\epsilon )$ to ${N}_{k}(b,\epsilon )$ is a $({k}_{0},{k}_{1})$-isomorphism for all $i\in M$.

**Remark**

**3.**

**Remark**

**4.**

**Proposition**

**2.**

- (1)
- In (10), if $i,j\in M$ and $i\ne j$, then ${N}_{{k}_{0}}({e}_{i},1)\cap {N}_{{k}_{0}}({e}_{j},1)$ is an empty set;
- (2)
- In (10), for any $i,j\in M$, ${N}_{{k}_{0}}({e}_{i},1)$ is ${k}_{0}$-isomorphic to ${N}_{{k}_{0}}({e}_{j},1)$.
- (3)
- In (10), for any $i,j\in M$ and $i\ne j$, ${N}_{{k}_{0}}({e}_{i},1)$ is not ${k}_{0}$-adjacent to ${N}_{{k}_{0}}({e}_{j},1)$.

**Proof.**

**Corollary**

**4.**

**Proof.**

- (1)
- By Remark 2(1), the given map is a $({k}_{0},{k}_{1})$-continuous map.
- (2)
- By Theorem 2, the given map is a surjection.
- (3)
- As mentioned in Remark 3(2), in Definition 7 we may take $\epsilon =1$. Then, the properties from Proposition 2(1)–(2) with the hypothesis imply the axioms (2)–(3) of Definition 7. Naively, we obtain that ${N}_{{k}_{0}}({e}_{i},1)\cap {N}_{{k}_{0}}({e}_{j},1)=\varnothing $ and further, the restriction of p on ${N}_{{k}_{0}}(e,1)$, ${p|}_{{N}_{{k}_{0}}(e,1)}:{N}_{{k}_{0}}(e,1)\to {N}_{{k}_{1}}(p(e),1)$, is a $({k}_{0},{k}_{1})$-isomorphism. ☐

**Remark**

**5.**

- (1)
- For some index set M, ${p}^{-1}({N}_{{k}_{1}}(b,1))={\cup}_{i\in M}{N}_{{k}_{0}}({e}_{i},1)$ with ${e}_{i}\in {p}^{-1}(b)$;
- (2)
- if $i,j\in M$ and $i\ne j$, then ${N}_{{k}_{0}}({e}_{i},1)\cap {N}_{{k}_{0}}({e}_{j},1)$ is an empty set; and
- (3)
- the restriction of p to ${N}_{{k}_{0}}({e}_{i},1)$ from ${N}_{k}({e}_{i},1)$ to ${N}_{k}(b,1)$ is a $({k}_{0},{k}_{1})$-isomorphism for all $i\in M$.

**Corollary**

**5.**

**Definition**

**8.**

**Remark**

**6.**

**Theorem**

**3.**

**Proposition**

**3.**

**Proof.**

**Remark**

**7.**

**Remark**

**8.**

## 6. Generalized Digital Wedges and Alignments of Fixed Point Sets of ${\mathit{SC}}_{{\mathit{k}}_{\mathbf{1}}}^{{\mathit{n}}_{\mathbf{1}},{\mathit{l}}_{\mathbf{1}}}\vee {\mathit{SC}}_{{\mathit{k}}_{\mathbf{2}}}^{{\mathit{n}}_{\mathbf{2}},{\mathit{l}}_{\mathbf{2}}}$, Where ${\mathit{n}}_{\mathbf{1}}\ne {\mathit{n}}_{\mathbf{2}}$ or ${\mathit{k}}_{\mathbf{1}}\ne {\mathit{k}}_{\mathbf{2}}$

**Definition**

**9.**

**Remark**

**9.**

**Example**

**4.**

**Definition**

**10.**

**Definition**

**11.**

**Lemma**

**2.**

**Theorem**

**4.**

**Proof.**

- (a)
- ${f|}_{S{C}_{26}^{3,5}}(x)=x$; or
- (b)
- ${f|}_{S{C}_{8}^{2,l}}(x)=x$; or
- (c)
- $f(S{C}_{8}^{2,l})\u228aS{C}_{8}^{2,l}$ and $f(S{C}_{26}^{3,5})\u228aS{C}_{26}^{3,5}$; or
- (d)
- f does not have any fixed point of it, where ${f|}_{X}$ means the restriction function f to the given set X.

**Example**

**5.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**6.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 7. Conclusions

## Funding

## Conflicts of Interest

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**Figure 1.**Configuration of several types of $S{C}_{k}^{n,l}$: (

**a**) $S{C}_{8}^{2,7}:={({x}_{i})}_{i\in {[0,6]}_{\mathbb{Z}}}$ [34], (

**b**) $S{C}_{8}^{2,9}:={({x}_{i})}_{i\in {[0,8]}_{\mathbb{Z}}}$ [34], (

**c**) $S{C}_{26}^{3,5}:={({x}_{i})}_{i\in {[0,4]}_{\mathbb{Z}}}$. Here, $n=3$, the underlying 3-dimensional lattice is shown as a dashed grid. The simple closed 26-curve consisting of 5 points appears as black dots labelled ${x}_{0}$ through ${x}_{4}$. Indeed, there are several shapes of $S{C}_{26}^{3,5}$ in ${\mathbb{Z}}^{3}$.

**Figure 2.**Configuration of local 8-isomorphisms of Definition 4 referred to in Example 2. (

**a**) ${p}_{1}:(X,8)\to S{C}_{8}^{2,12}$. (

**b**) ${p}_{2}:(Y,8)\to S{C}_{8}^{2,13}$.

**Figure 3.**Explanation of the map h referred to in the proof of Corollary 3, where $(Y,8)$ is a portion of ${\mathbb{Z}}^{2}$ with 8-adjacency.

**Figure 4.**(

**a**) Configuration of a $(26,18)$-continuous surjection f which is a local $(26,18)$-isomorphism mentioned in Example 3(1). (

**b**) Configuration of a $(2,26)$-continuous surjection which is not a local $(2,26)$-isomorphism stated in Example 3(2). (

**c**) Configuration of an $(8,26)$-surjection $p:(X,8)\to S{C}_{8}^{2,7}:={({y}_{i})}_{i\in {[0,6]}_{\mathbb{Z}}}$ which is not a local $(8,26)$-isomorphism referred to in Example 3(3), where X consists of six elements.

**Figure 5.**Configuration of an $(8,26)$-continuous surjection which is not a local $(8,26)$-isomorphism referred to in (18), where E is a portion of an infinite set (or an infinite fence set with 8-adjacency) in (17).

**Figure 6.**Configuration of digital wedges of two simple closed k-curves, $k\in \{8,18,26\}$, $S{C}_{26}^{3,5}:={({x}_{i})}_{i\in {[0,4]}_{\mathbb{Z}}}$ and $S{C}_{18}^{3,6}:={({y}_{i})}_{i\in {[0,5]}_{\mathbb{Z}}}$. (

**a**) $S{C}_{26}^{3,5}\vee S{C}_{26}^{3,5}$. Here, $n=3$, the underlying 3-dimensional lattice is shown as a dashed grid. The digital wedge generated by the two simple closed 26-curve consisting of 5 points appears as black dots labelled ${x}_{0}$ through ${x}_{8}$ with $S{C}_{26}^{3,5}\cap S{C}_{26}^{3,5}=\left\{{x}_{0}\right\}$. (

**b**) $(S{C}_{26}^{3,5}\vee S{C}_{8}^{2,6},26)=(S{C}_{26}^{3,5}\vee S{C}_{18}^{3,6},26)$. Here, $n=3$, the underlying 3-dimensional lattice is shown as a dashed grid. The digital wedge generated by both the simple closed 26-curve and the simple closed 18- or 8-curve appears as black dots labelled ${x}_{0}$ through ${x}_{4}$ and ${y}_{1}$ through ${y}_{5}$ with $S{C}_{26}^{3,5}\cap S{C}_{18}^{3,6}=\left\{{x}_{0}\right\}$

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Han, S.-E.
The Most Refined Axiom for a Digital Covering Space and Its Utilities. *Mathematics* **2020**, *8*, 1868.
https://doi.org/10.3390/math8111868

**AMA Style**

Han S-E.
The Most Refined Axiom for a Digital Covering Space and Its Utilities. *Mathematics*. 2020; 8(11):1868.
https://doi.org/10.3390/math8111868

**Chicago/Turabian Style**

Han, Sang-Eon.
2020. "The Most Refined Axiom for a Digital Covering Space and Its Utilities" *Mathematics* 8, no. 11: 1868.
https://doi.org/10.3390/math8111868