# The Sequential and Contractible Topological Embeddings of Functional Groups

## Abstract

**:**

## 1. Introduction

#### 1.1. Topological Embeddings

#### 1.2. Motivation

## 2. Preliminary Concepts

## 3. Decomposition and Functional Groups Embeddings: Concepts and Definitions

#### 3.1. Embeddable Topological Decomposition of Groups

#### 3.1.1. Definition of $G-$ Partition

#### 3.1.2. Topological Decomposition and Fiber

**Remark**

**1.**

#### 3.1.3. Decomposed Group Embedding

#### 3.2. Functional Groups and Topological Embeddings

#### 3.2.1. Functional Group

#### 3.2.2. Functional Group Embeddings

#### 3.2.3. Embedded Group Homeomorphism

#### 3.2.4. Jordan Curve Components

#### 3.2.5. Sequential Embeddings

#### 3.2.6. Component Boundary Embeddings

#### 3.2.7. Uniform Contraction

## 4. Main Results

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**2.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Remark**

**3.**

**Theorem**

**8.**

**Proof.**

## 5. Comparative Analysis

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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Group Structures | Homeomorphism | Topological Decomposability | Sequentially Embeddable | Contraction | One-point Compactification |
---|---|---|---|---|---|

${G}_{i}=({S}_{i},{\ast}_{i},{f}_{i})$ | ${S}^{1}$ | Yes | Yes | Yes | Possible |

${G}_{X}=(X,{\ast}_{X})$ | ${G}_{Y}=(Y,{\ast}_{Y})$ | Yes | No | No | Possible |

Decomposition/Embeddings | Uniqueness | Global Continuity | Local Continuity | Identity | Separability |
---|---|---|---|---|---|

Semidirect product | No | Yes | Yes | Shared | No |

Direct product | Yes | No | Yes | Distinct | Yes |

Topological decomposition | Yes | No | Yes | Distinct | Yes |

Functional group embeddings | Yes | No | Yes | Distinct | Monotone class components: No, otherwise: Yes |

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Bagchi, S.
The Sequential and Contractible Topological Embeddings of Functional Groups. *Symmetry* **2020**, *12*, 789.
https://doi.org/10.3390/sym12050789

**AMA Style**

Bagchi S.
The Sequential and Contractible Topological Embeddings of Functional Groups. *Symmetry*. 2020; 12(5):789.
https://doi.org/10.3390/sym12050789

**Chicago/Turabian Style**

Bagchi, Susmit.
2020. "The Sequential and Contractible Topological Embeddings of Functional Groups" *Symmetry* 12, no. 5: 789.
https://doi.org/10.3390/sym12050789