Meet the editor

Dr. Francisco Bulnes, Ph.D., is IINAMEI Director, Mathematics Research Centre, Mexico. He is a member of various international committees of science and serves as a reviewer and editor for British and American journals of mathematics and physics. He is head of the Research Department, GI-TESCHA. He has published more than 100 papers and several books in mathematics and physics. Dr. Bulnes has many theories, theorems, and

math objects to his credit. He has received various honors and awards from universities as well as governmental and non-governmental organizations. He received the Doctor Honoris Causa in Education Philosophy and is a Peace Ambassador for ODAEE in Frankfurt, Germany. He is also a distinguished member of the Czech Republic Mathematics Society (JCFM). He obtained two post-doctorates in Mathematics in Cuba and Russia. His research interests include electronics, microelectronics, and spintronics.

Contents

**Section 1**

Last Frontiers *by Francisco Bulnes*

**Section 2**

**Section 3**

for Cycloalkenes *by Yasuhiko Kamiyama*

*by Lev Bukovský*

**Section 4**

**Preface XI**

Introduction **1**

**Chapter 1 3**

Special Topological Sets and Their Continous Applications **11**

**Chapter 2 13**

**Chapter 3 25**

Cobordisms, Coverings and Topological Sheepers **37**

**Chapter 4 39**

**Chapter 5 61**

Combinatoral Topology and Descompoibilities to Shellability **73**

**Chapter 6 75**

Vertex Decomposability of Path Complexes and Stanley's Conjectures

Introductory Chapter: The Topology from Classic Studies until Its

More Functions Associated with Neutrosophic gsα\*- Closed Sets

The Topology of the Configuration Space of a Mathematical Model

in Neutrosophic Topological Spaces *by P. Anbarasi Rodrigo and S. Maheswari*

4-Dimensional Canards with Brownian Motion *by Shuya Kanagawa and Kiyoyuki Tchizawa*

Covers and Properties of Families of Real Functions

*by Seyed Mohammad Ajdani and Francisco Bulnes*

## Contents



Preface

Topology, originally called "analysis situs," explores the properties and relations between geometric objects (or objects with certain set properties) in an ambient space. Topological invariants, which are properties that are preserved under a homeomorphism, are of great importance to the field of topology since many of them characterize geometric spaces and dynamical systems and define metrizability in spaces. Over six sections, this book discusses several types of topologies, including algebraic topology, differential topology, and symplectic topology, as well as topological spaces and manifolds, dimension theory, and the general study of topology. Modern topology developments focus on items and contents such as homotopy, cohomology, non-commutative rings, cobordisms, Lindelöf spaces, projective manifolds, connectivity, topos, and singular manifolds, all of which play a role in the development of deferent string theories and the topology of quantum field theories to the conceptual precision and understanding of the Universe. Also, in artificial intelligence and numerical simulation, topology is fundamental for defining spaces with adequate metrizability in dynamical systems and the design of artificial intelligence units of advanced automatons and the approach of androids

**Dr. Francisco Bulnes, Ph.D, PostDocs, Doctor H. C., HonDSc, ZbMath, MathSci**

Research Department in Mathematics and Engineering,

Professor, IINAMEI,

TESCHA, Chalco, Mexico

to human behavior.

*by Sanjay Kumar Singh and Punam Gupta*
