# On the Topological Structure and Properties of Multidimensional (C, R) Space

## Abstract

**:**

## 1. Introduction

#### 1.1. Topology of Complex Analytic Spaces

#### 1.2. Topology of Normed Linear Spaces

#### 1.3. Motivation and Contributions

## 2. Preliminary Concepts

## 3. The Multidimensional $(\mathit{C}\mathbf{,}\mathit{R})$ Space

#### 3.1. Topological Projections in $(C,R)$ Space

#### 3.2. Non-Uniform Scaling in $(C,R)$ Space

#### 3.3. Projective Norms

#### 3.4. Quasinorm in Topological $(C,R)$ Space

**Proposition**

**1.**

**Proof.**

#### 3.5. Open Cylindrical Basis

**Remark**

**1.**

#### 3.6. Finite Translation in Topological $(C,R)$ Space

**Remark**

**2.**

#### 3.7. Cylindrical Symmetry of Continuous Function

## 4. Main Results

#### 4.1. Topological Group in (C, R) Space

#### 4.1.1. Condition for Associative $\ast $

#### 4.1.2. Condition for Identity Element

**Remark**

**3.**

#### 4.1.3. Existence of Inverse Element

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 4.2. Analysis of Topological Properties

**Theorem**

**3.**

**Proof.**

**Remark**

**4.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**7.**

**Proof.**

**Lemma**

**2.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Bagchi, S.
On the Topological Structure and Properties of Multidimensional (*C*, *R*) Space. *Symmetry* **2020**, *12*, 1542.
https://doi.org/10.3390/sym12091542

**AMA Style**

Bagchi S.
On the Topological Structure and Properties of Multidimensional (*C*, *R*) Space. *Symmetry*. 2020; 12(9):1542.
https://doi.org/10.3390/sym12091542

**Chicago/Turabian Style**

Bagchi, Susmit.
2020. "On the Topological Structure and Properties of Multidimensional (*C*, *R*) Space" *Symmetry* 12, no. 9: 1542.
https://doi.org/10.3390/sym12091542