# The Stereographic Projection in Topological Modules

*Axioms:*Geometry and Topology)

## Abstract

**:**

## 1. Introduction

## 2. Methodology

## 3. Results

#### 3.1. Fit Bodies

**Definition 1**(Fit body).

**Proposition 1.**

**Proof.**

**Example 1.**

**Theorem 1.**

**Proof.**

#### 3.2. Minkowski Bodies

**Definition 2**(Minkowski functional).

**Proposition 2.**

**Proof.**

**Definition 3**(Minkowski body).

**Example 2.**

**Theorem 2.**

**Proof.**

#### 3.3. Exposed Faces

**Lemma 1.**

**Proof.**

**Definition 4**(Exposed face).

**Theorem 3.**

**Proof.**

**Corollary 1.**

**Proof.**

#### 3.4. The Stereographic Projection

**Definition 5**(Stereographic projection).

**Remark 1.**

**Definition 6**(Strongly rotund point).

- ${m}^{*}\left(A\right)$ is relatively compact in R.
- ${\left({m}^{*}\right)}^{-1}\left(\left\{s\right\}\right)\cap \mathrm{int}\left(A\right)=\u2300$.
- ${\left({m}^{*}\right)}^{-1}\left(\left\{s\right\}\right)\cap \mathrm{bd}\left(A\right)=\left\{a\right\}$.
- $s+\mathrm{cl}\left({m}^{*}(\mathrm{bd}\left(A\right)\backslash \{-a\})\right)\backslash \{-s\}\subseteq \mathcal{U}\left(R\right)$.
- ${\mathrm{st}}_{\mathcal{U}}(-a,b)\cap \mathrm{bd}\left(A\right)=\{-a,b\}$ for all $b\in \mathrm{bd}\left(A\right)$.
- ${\mathrm{st}}_{\mathcal{U}}(-a,c)\cap (\mathrm{bd}\left(A\right)\backslash \{-a\})\ne \u2300$ for all $c\in {\left({m}^{*}\right)}^{-1}\left(\left\{s\right\}\right)$.
- If ${\left({c}_{j}\right)}_{j\in J}\subseteq \mathrm{bd}\left(A\right)\backslash \{-a\}$ is a net converging to $-a$, then ${\left({({m}^{*}\left({c}_{j}\right)+s)}^{-1}({c}_{j}+a)\right)}_{j\in J}$ is not convergent.

**Remark 2.**

- $s\ne 0$ because $0\in \mathrm{int}\left(A\right)$ and ${\left({m}^{*}\right)}^{-1}\left(\left\{s\right\}\right)\cap \mathrm{int}\left(A\right)=\u2300$. Additionally, if $b\in \mathrm{bd}\left(A\right)\backslash \{-a\}$, then ${m}^{*}\left(b\right)\ne -s$ because otherwise we conclude that $b=-a$ in view of the fact that ${\left({m}^{*}\right)}^{-1}\left(\left\{s\right\}\right)\cap \mathrm{bd}\left(A\right)=\left\{a\right\}$. As a consequence, $s+{m}^{*}(\mathrm{bd}\left(A\right)\backslash \{-a\})\subseteq \mathcal{U}\left(R\right)$. For these reasons, the condition $s+\mathrm{cl}\left({m}^{*}(\mathrm{bd}\left(A\right)\backslash \{-a\})\right)\backslash \{-s\}\subseteq \mathcal{U}\left(R\right)$ is well-imposed in Definition 6.
- If $\mathrm{char}\left(R\right)\ne 2$, then $-a\notin {\left({m}^{*}\right)}^{-1}\left(\left\{s\right\}\right)$ or equivalently $-a\ne a$. Indeed, if $-a\in {\left({m}^{*}\right)}^{-1}\left(\left\{s\right\}\right)$, then $-s=s$, hence $(1+1)s=0$, so $1+1=0$ because s is invertible, contradicting that $\mathrm{char}\left(R\right)\ne 2$.

**Lemma 2.**

**Proof.**

**Lemma 3.**

**Proof.**

**Theorem 4.**

**Proof.**

## 4. Discussion

**Theorem 5.**

**Proof.**

**Example 3.**

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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García-Pacheco, F.J.
The Stereographic Projection in Topological Modules. *Axioms* **2023**, *12*, 225.
https://doi.org/10.3390/axioms12020225

**AMA Style**

García-Pacheco FJ.
The Stereographic Projection in Topological Modules. *Axioms*. 2023; 12(2):225.
https://doi.org/10.3390/axioms12020225

**Chicago/Turabian Style**

García-Pacheco, Francisco Javier.
2023. "The Stereographic Projection in Topological Modules" *Axioms* 12, no. 2: 225.
https://doi.org/10.3390/axioms12020225