# Pre-Metric Spaces Along with Different Types of Triangle Inequalities

## Abstract

**:**

## 1. Introduction

- for any $x,y\in M$, $d(x,y)=0$ implies $x=y$;
- (self-distance condition) for any $x\in M$, $d(x,x)=0$;
- (symmetric condition) for any $x,y\in M$, $d(x,y)=d(y,x)$;
- (triangle inequality) for any $x,y,z\in M$, $d(x,z)\le d(x,y)+d(y,z)$.

- for any $x,y\in M$, $d(x,y)=0$ if and only if $x=y$;
- for any $x,y,z\in M$, $d(x,z)\le d(x,y)+d(y,z)$.

- for any $x,y\in M$, $d(x,y)=0=d(y,x)$ if and only if $x=y$;
- for any $x,y,z\in M$, $d(x,z)\le d(x,y)+d(y,z)$.

- for any $x,y\in M$, $d(x,y)=0$ if and only if $x=y$;
- for any $x,y\in M$, $d(x,y)=d(y,x)$.

- for any $x,y\in M$, $x=y$ if and only if $d(x,x)=d(x,y)=d(y,y)$;
- for any $x,y\in M$, $d(x,x)\le d(x,y)$;
- for any $x,y\in M$, $d(x,y)=d(y,x)$.
- for any $x,y,z\in M$, $d(x,z)\le d(x,y)+d(y,z)-d(y,y)$.

## 2. Definitions and Properties

**Definition**

**1.**

- We say that d satisfies the ⋈-triangle inequality if and only if the following inequality is satisfied:$$d(x,y)+d(y,z)\ge d(x,z)\text{}for\text{}all\text{}x,y,z\in X.$$
- We say that d satisfies the ▹-triangle inequality if and only if the following inequality is satisfied:$$d(x,y)+d(z,y)\ge d(x,z)\text{}for\text{}all\text{}x,y,z\in X.$$
- We say that d satisfies the ◃-triangle inequality if and only if the following inequality is satisfied:$$d(y,x)+d(y,z)\ge d(x,z)\text{}for\text{}all\text{}x,y,z\in X.$$
- We say that d satisfies the ⋄-triangle inequality if and only if the following inequality is satisfied:$$d(y,x)+d(z,y)\ge d(x,z)\text{}for\text{}all\text{}x,y,z\in X.$$

**Example**

**1.**

**Example**

**2.**

**Definition**

**2.**

**Example**

**3.**

**Remark**

**1.**

**Proposition**

**1.**

- $d(x,x)=0$ for all $x\in X$;
- d satisfies the ▹-triangle inequality or the ◃-triangle inequality or the ⋄-triangle inequality.

**Proof.**

**Remark**

**2.**

## 3. ${\mathit{T}}_{\mathbf{1}}$-Space

**Definition**

**3.**

**Proposition**

**2.**

- (i)
- Given any $x\in X$, we have the following properties.
- Suppose that $d(x,x)=0$. Then $x\in {B}^{\u25c3}(x;r)\in {\mathcal{B}}^{\u25c3}$ and $x\in {B}^{\u25b9}(x;r)\in {\mathcal{B}}^{\u25b9}$ for all $r>0$.
- Suppose that $x\in {B}^{\u25c3}(x;r)$ for all $r>0$, or that $x\in {B}^{\u25b9}(x;r)$ for all $r>0$. Then $d(x,x)=0$.

- (ii)
- If $x\ne y$, then there exist ${r}_{1}>0$ and ${r}_{2}>0$ such that $y\notin {B}^{\u25c3}(x;{r}_{1})$ and $y\notin {B}^{\u25b9}(x;{r}_{2})$.
- (iii)
- For each $x\in X$, we have the following properties.
- Given any ${B}^{\u25c3}(x;r)\in {\mathcal{B}}^{\u25c3}$, there exists $n\in \mathbb{N}$ such that ${B}^{\u25c3}(x;\frac{1}{n})\subseteq {B}^{\u25c3}(x;r)$.
- Given any ${B}^{\u25b9}(x;r)\in {\mathcal{B}}^{\u25b9}$, there exists $n\in \mathbb{N}$ such that ${B}^{\u25b9}(x;\frac{1}{n})\subseteq {B}^{\u25b9}(x;r)$.

**Proof.**

**Proposition**

**3.**

- (i)
- Suppose that d satisfies the ⋈-triangle inequality.
- Given any $y\in {B}^{\u25c3}(x;r)$, there exists $\overline{r}>0$ such that ${B}^{\u25c3}(y;\overline{r})\subseteq {B}^{\u25c3}(x;r)$.
- Given any $y\in {B}^{\u25b9}(x;r)$, there exists $\overline{r}>0$ such that ${B}^{\u25b9}(y;\overline{r})\subseteq {B}^{\u25b9}(x;r)$.

- (ii)
- Suppose that d satisfies the ▹-triangle inequality. Given any $y\in {B}^{\u25c3}(x;r)$, there exists $\overline{r}>0$ such that ${B}^{\u25b9}(y;\overline{r})\subseteq {B}^{\u25c3}(x;r)$ and ${B}^{\u25b9}(y;\overline{r})\subseteq {B}^{\u25b9}(x;r)$.
- (iii)
- Suppose that d satisfies the ◃-triangle inequality. Given any $y\in {B}^{\u25b9}(x;r)$, there exists $\overline{r}>0$ such that ${B}^{\u25c3}(y;\overline{r})\subseteq {B}^{\u25b9}(x;r)$ and ${B}^{\u25c3}(y;\overline{r})\subseteq {B}^{\u25c3}(x;r)$.
- (iv)
- Suppose that d satisfies the ⋄-triangle inequality.
- Given any $y\in {B}^{\u25c3}(x;r)$, there exists $\overline{r}>0$ such that ${B}^{\u25c3}(y;\overline{r})\subseteq {B}^{\u25b9}(x;r)$.
- Given any $y\in {B}^{\u25b9}(x;r)$, there exists $\overline{r}>0$ such that ${B}^{\u25b9}(y;\overline{r})\subseteq {B}^{\u25c3}(x;r)$.

- (v)
- Suppose that d satisfies the ▹-triangle inequality and the ◃-triangle inequality.
- Given any $y\in {B}^{\u25c3}(x;r)$, there exists $\overline{r}>0$ such that ${B}^{\u25c3}(y;\overline{r})\subseteq {B}^{\u25c3}(x;r)$.
- Given any $y\in {B}^{\u25b9}(x;r)$, there exists $\overline{r}>0$ such that ${B}^{\u25b9}(y;\overline{r})\subseteq {B}^{\u25b9}(x;r)$.

**Proof.**

**Proposition**

**4.**

- (i)
- Suppose that d satisfies the ⋈-triangle inequality.
- If $x\in {B}^{\u25c3}({x}_{1},{r}_{1})\cap {B}^{\u25c3}({x}_{2},{r}_{2})$, then there exists ${r}_{3}>0$ such that$${B}^{\u25c3}(x,{r}_{3})\subseteq {B}^{\u25c3}({x}_{1},{r}_{1})\cap {B}^{\u25c3}({x}_{2},{r}_{2}).$$
- If $x\in {B}^{\u25b9}({x}_{1},{r}_{1})\cap {B}^{\u25b9}({x}_{2},{r}_{2})$, then there exists ${r}_{3}>0$ such that$${B}^{\u25b9}(x,{r}_{3})\subseteq {B}^{\u25b9}({x}_{1},{r}_{1})\cap {B}^{\u25b9}({x}_{2},{r}_{2}).$$

- (ii)
- Suppose that d satisfies the ▹-triangle inequality. If $x\in {B}^{\u25c3}({x}_{1},{r}_{1})\cap {B}^{\u25c3}({x}_{2},{r}_{2})$, then there exists ${r}_{3}>0$ such that$${B}^{\u25b9}(x,{r}_{3})\subseteq {B}^{\u25c3}({x}_{1},{r}_{1})\cap {B}^{\u25c3}({x}_{2},{r}_{2})\text{}and\text{}{B}^{\u25b9}(x,{r}_{3})\subseteq {B}^{\u25b9}({x}_{1},{r}_{1})\cap {B}^{\u25b9}({x}_{2},{r}_{2}).$$
- (iii)
- Suppose that d satisfies the ◃-triangle inequality. If $x\in {B}^{\u25b9}({x}_{1},{r}_{1})\cap {B}^{\u25b9}({x}_{2},{r}_{2})$, then there exists ${r}_{3}>0$ such that$${B}^{\u25c3}(x,{r}_{3})\subseteq {B}^{\u25b9}({x}_{1},{r}_{1})\cap {B}^{\u25b9}({x}_{2},{r}_{2})\text{}and\text{}{B}^{\u25c3}(x,{r}_{3})\subseteq {B}^{\u25c3}({x}_{1},{r}_{1})\cap {B}^{\u25c3}({x}_{2},{r}_{2}).$$
- (iv)
- Suppose that d satisfies the ⋄-triangle inequality.
- If $x\in {B}^{\u25c3}({x}_{1},{r}_{1})\cap {B}^{\u25c3}({x}_{2},{r}_{2})$, then there exists ${r}_{3}>0$ such that$${B}^{\u25c3}(x,{r}_{3})\subseteq {B}^{\u25b9}({x}_{1},{r}_{1})\cap {B}^{\u25b9}({x}_{2},{r}_{2}).$$
- If $x\in {B}^{\u25b9}({x}_{1},{r}_{1})\cap {B}^{\u25b9}({x}_{2},{r}_{2})$, then there exists ${r}_{3}>0$ such that$${B}^{\u25b9}(x,{r}_{3})\subseteq {B}^{\u25c3}({x}_{1},{r}_{1})\cap {B}^{\u25c3}({x}_{2},{r}_{2}).$$

- (v)
- Suppose that d satisfies the ▹-triangle inequality and the ◃-triangle inequality. We have the following inclusions.
- If $x\in {B}^{\u25c3}({x}_{1},{r}_{1})\cap {B}^{\u25c3}({x}_{2},{r}_{2})$, then there exists ${r}_{4}>0$ such that$${B}^{\u25c3}(x,{r}_{4})\subseteq {B}^{\u25c3}({x}_{1},{r}_{1})\cap {B}^{\u25c3}({x}_{2},{r}_{2}).$$
- If $x\in {B}^{\u25b9}({x}_{1},{r}_{1})\cap {B}^{\u25b9}({x}_{2},{r}_{2})$, then there exists ${r}_{4}>0$ such that$${B}^{\u25b9}(x,{r}_{4})\subseteq {B}^{\u25b9}({x}_{1},{r}_{1})\cap {B}^{\u25b9}({x}_{2},{r}_{2}).$$

**Proof.**

**Proposition**

**5.**

- (i)
- Suppose that d satisfies the ⋈-triangle inequality or the ⋄-triangle inequality. Then ${B}^{\u25c3}(x;r)\cap {B}^{\u25b9}(y;r)=\varnothing $ and ${B}^{\u25b9}(x;r)\cap {B}^{\u25c3}(y;r)=\varnothing $ for some $r>0$.
- (ii)
- Suppose that d satisfies the ▹-triangle inequality. Then ${B}^{\u25c3}(x;r)\cap {B}^{\u25c3}(y;r)=\varnothing $ for some $r>0$.
- (iii)
- Suppose that d satisfies the ◃-triangle inequality. Then ${B}^{\u25b9}(x;r)\cap {B}^{\u25b9}(y;r)=\varnothing $ for some $r>0$.

**Proof.**

- Suppose that d satisfies the ▹-triangle inequality. Let $r\le d(x,y)/2$. We are going to prove ${B}^{\u25c3}(x;r)\cap {B}^{\u25c3}(y;r)=\varnothing $ by contradiction. Suppose that $z\in {B}^{\u25c3}(x;r)\cap {B}^{\u25c3}(y;r)$. Since d satisfies the ▹-triangle inequality, it follows that$$d(x,y)\le d(x,z)+d(y,z)<r+r=2r\le d(x,y),$$
- Suppose that d satisfies the ⋈-triangle inequality. Let $r\le d(x,y)/2$. For $z\in {B}^{\u25c3}(x;r)\cap {B}^{\u25b9}(y;r)$, it follows that$$d(x,y)\le d(x,z)+d(z,y)<r+r=2r\le d(x,y),$$which is a contradiction. On the other hand, let $r\le d(y,x)/2$, for $z\in {B}^{\u25b9}(x;r)\cap {B}^{\u25c3}(y;r)$, it follows that$$d(y,x)\le d(y,z)+d(z,x)<r+r=2r\le d(y,x),$$

**Theorem**

**1.**

- Assume additionally that $d(x,x)=0$ for all $x\in X$, or that $x\in {B}^{\u25c3}(x;r)$ for all $x\in X$ and $r>0$. Then $(X,{\tau}^{\u25c3})$ is a ${T}_{1}$-space such that ${\mathcal{B}}^{\u25c3}$ is a base for the topology ${\tau}^{\u25c3}$.
- Assume additionally that $d(x,x)=0$ for all $x\in X$, or that $x\in {B}^{\u25b9}(x;r)$ for all $x\in X$ and $r>0$. Then $(X,{\tau}^{\u25b9})$ is a ${T}_{1}$-space such that ${\mathcal{B}}^{\u25b9}$ is a base for the topology ${\tau}^{\u25b9}$.

**Proof.**

**Example**

**4.**

## 4. Limits in Pre-Metric Space

**Definition**

**4.**

- We write ${x}_{n}\stackrel{{d}^{\u25b9}}{\u27f6}x$ as $n\to \infty $ if and only if $d({x}_{n},x)\to 0$ as $n\to \infty $.
- We write ${x}_{n}\stackrel{{d}^{\u25c3}}{\u27f6}x$ as $n\to \infty $ if and only if $d(x,{x}_{n})\to 0$ as $n\to \infty $.
- We write ${x}_{n}\stackrel{d}{\u27f6}x$ as $n\to \infty $ if and only if$$\underset{n\to \infty}{lim}d({x}_{n},x)=\underset{n\to \infty}{lim}d(x,{x}_{n})=0.$$

**Proposition**

**6.**

- (i)
- Suppose that d satisfies the ⋈-triangle inequality or ⋄-triangle inequality. If ${x}_{n}\stackrel{{d}^{\u25c3}}{\u27f6}x$ and ${x}_{n}\stackrel{{d}^{\u25b9}}{\u27f6}y$, then $x=y$.
- (ii)
- Suppose that d satisfies the ◃-triangle inequality. If ${x}_{n}\stackrel{{d}^{\u25b9}}{\u27f6}x$ and ${x}_{n}\stackrel{{d}^{\u25b9}}{\u27f6}y$, then $x=y$. In other words, the ${d}^{\u25b9}$-limit is unique.
- (iii)
- Suppose that d satisfies the ▹-triangle inequality. If ${x}_{n}\stackrel{{d}^{\u25c3}}{\u27f6}x$ and ${x}_{n}\stackrel{{d}^{\u25c3}}{\u27f6}y$, then $x=y$. In other words, the ${d}^{\u25c3}$-limit is unique.

**Proof.**

**Proposition**

**7.**

- (i)
- Let ${\tau}^{\u25b9}$ be the topology defined by (1) in Theorem 1, and let ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ be a sequence in X. Then ${x}_{n}\stackrel{{\tau}^{\u25b9}}{\u27f6}x$ as $n\to \infty $ if and only if ${x}_{n}\stackrel{{d}^{\u25b9}}{\u27f6}x$ as $n\to \infty $.
- (ii)
- Let ${\tau}^{\u25c3}$ be the topology defined by (2) in Theorem 1, and let ${\left\{{x}_{n}\right\}}_{n=1}^{\infty}$ be a sequence in X. Then ${x}_{n}\stackrel{{\tau}^{\u25c3}}{\u27f6}x$ as $n\to \infty $ if and only if ${x}_{n}\stackrel{{d}^{\u25c3}}{\u27f6}x$ as $n\to \infty $.

**Proof.**

**Proposition**

**8.**

- ${\overline{B}}^{\u25c3}(x;r)$ is ${\tau}^{\u25b9}$-closed. In other words, we have ${\tau}^{\u25b9}-\mathrm{cl}({\overline{B}}^{\u25c3}(x;r))={\overline{B}}^{\u25c3}(x;r)$.
- ${\overline{B}}^{\u25b9}(x;r)$ is ${\tau}^{\u25c3}$-closed. In other words, we have ${\tau}^{\u25c3}-\mathrm{cl}({\overline{B}}^{\u25b9}(x;r))={\overline{B}}^{\u25b9}(x;r)$.

**Proof.**

**Proposition**

**9.**

- d satisfies the ▹-triangle inequality and the ◃-triangle inequality simultaneously.
- $d(x,x)=0$ for all $x\in X$.

**Proof.**

**Remark**

**3.**

## Conflicts of Interest

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**MDPI and ACS Style**

Wu, H.-C.
Pre-Metric Spaces Along with Different Types of Triangle Inequalities. *Axioms* **2018**, *7*, 34.
https://doi.org/10.3390/axioms7020034

**AMA Style**

Wu H-C.
Pre-Metric Spaces Along with Different Types of Triangle Inequalities. *Axioms*. 2018; 7(2):34.
https://doi.org/10.3390/axioms7020034

**Chicago/Turabian Style**

Wu, Hsien-Chung.
2018. "Pre-Metric Spaces Along with Different Types of Triangle Inequalities" *Axioms* 7, no. 2: 34.
https://doi.org/10.3390/axioms7020034