# Varieties of Coarse Spaces

## Abstract

**:**

## 1. Introduction

- Each $\epsilon \in \mathcal{E}$ contains the diagonal ${\u25b5}_{X}$, ${\u25b5}_{X}=\{(x,x):x\in X\}$;
- If $\epsilon ,\delta \in \mathcal{E}$ then $\epsilon \circ \delta \in \mathcal{E}$ and ${\epsilon}^{-1}\in \mathcal{E}$, where $\epsilon \circ \delta =\left\{\right(x,y):\exists z((x,z)\in \epsilon ,(z,y)\in \delta \left)\right\},$ and $\phantom{\rule{4pt}{0ex}}{\epsilon}^{-1}=\{(y,x):(x,y)\in \epsilon \}$;
- And if $\epsilon \in \mathcal{E}$ and ${\u25b3}_{X}\subseteq {\epsilon}^{\prime}\subseteq \epsilon $ then ${\epsilon}^{\prime}\in \mathcal{E}$.

**connected**.

## 2. Results

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 3. Comments

## Conflicts of Interest

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Protasov, I. Varieties of Coarse Spaces. *Axioms* **2018**, *7*, 32.
https://doi.org/10.3390/axioms7020032

**AMA Style**

Protasov I. Varieties of Coarse Spaces. *Axioms*. 2018; 7(2):32.
https://doi.org/10.3390/axioms7020032

**Chicago/Turabian Style**

Protasov, Igor. 2018. "Varieties of Coarse Spaces" *Axioms* 7, no. 2: 32.
https://doi.org/10.3390/axioms7020032