# Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

## Abstract

**:**

## 1. Introduction

#### 1.1. Homotopy and Decomposition

#### 1.2. Motivation

## 2. Preliminary Concepts

## 3. Connected Fundamental Groups and Decomposed Homotopy

#### 3.1. Topological Path Embedding

#### 3.2. Homeomorphic Embedding of Curve

#### 3.3. Connected Fundamental Groups

**Remark**

**1:**

#### 3.4. Decomposed Homotopy Loop

**Remark**

**2:**

#### 3.5. Decomposed Quotient Topology

#### 3.6. Cyclic Generator and Cycle Number

**Remark**

**3:**

## 4. Main Results

**Theorem**

**1:**

**Proof:**

**Example**

**1:**

**Corollary**

**1:**

**Proof:**

**Remark**

**4:**

**Theorem**

**2:**

**Proof:**

**Theorem**

**3:**

**Proof:**

**Remark**

**5:**

**Corollary**

**2:**

**Proof:**

**Theorem**

**4:**

**Proof:**

**Lemma**

**1:**

**Proof:**

**Theorem**

**5:**

**Proof:**

**Remark**

**6:**

**Theorem**

**6:**

**Proof:**

**Theorem**

**7:**

**Proof:**

## 5. Discussion: Relation to Sierpinski Space

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Space/Set | Extension | Topology | Group | Connectedness | Compact | Sequentially Complete (Non-Zero Sequence) | Non-Zero Sequence |
---|---|---|---|---|---|---|---|

Sierpinski | $\{\varphi \}$ | Sierpinski topology, ${\tau}_{SP}$ | Trivial | Yes | Compactible | Always | Unique cluster-point |

Decomposed homotopy quotient | $E$ | ${\tau}_{SP\gamma}={\tau}_{SP}\cup E$ | Non-trivial | Yes | Compactible | Not always | Not necessarily unique cluster-point |

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Bagchi, S.
Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces. *Symmetry* **2020**, *12*, 1039.
https://doi.org/10.3390/sym12061039

**AMA Style**

Bagchi S.
Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces. *Symmetry*. 2020; 12(6):1039.
https://doi.org/10.3390/sym12061039

**Chicago/Turabian Style**

Bagchi, Susmit.
2020. "Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces" *Symmetry* 12, no. 6: 1039.
https://doi.org/10.3390/sym12061039