Neutro Geometric Topology and Its Examples

Introduction

Topology made to study the surfaces of different objects whose properties remains invariant under homeomorphism. It is useful to study those objects whose geometrical properties are preserved under continuous deformation like stretching and bending. Due to which, it is known as “rubber-sheet geometry”. Topology can be regarded as a new branch of mathematics and it has some other subbranches like general topology (Sierpinski, 2020), geometric topology (Martelli, 2016), combinatorial topology (Weisstein), differential topology (Brocker& Janich, 1982), algebraic topology (Munkers, 2018), etc. On the other hand, simple geometry, or Euclidean geometry is the study of the shapes, sizes, and relative positions of 2D and 3D objects, based on axioms, postulates, and theorems. Topology and geometry seem to be akin to one another in many areas. There are many instances where geometric properties of objects and its study requires other things (attributes) like color, smell, melting point, etc. To understand these instances precisely the study of their topological properties is required. In this way, the topology helps in building the relationship among the physical and logical properties of an object. The above discussion shows that, geometry cannot answer many questions among the pair of objects and its instances which can be answered via topology. It is one of the reasons which gave flexibility to apply topology in numerous fields of science and technology (Adhikari, 2016). Cheng-Ming, (1985) extended the classical topology by introducing the fuzzy topological spaces. Sostak et al. (1996) studied the basic structures of fuzzy topology which is studied via Rough set theory base topological space by Lashin et al. (2005). Sarkar (1981) defined the L-fuzzy topological spaces (Sarkar, 1981) followed by Mondal and Samanta (2011) et al. presented topology via interval-valued intuitionistic fuzzy sets (Mondal & Samanta, 2001). Hur et al. (2014) introduced the intuitionistic fuzzy topological spaces for dealing the hesitant uncertainty. Recently, Pythagorean fuzzy topological spaces is introduced by Olgun et al. (2019) whereas Lee et al. (2022) introduced the hesitant fuzzy topological spaces for dealing with the hesitant or indeterminacy part. It gives a way to introduce a new geometry for dealing with the uncertainty and indeterminacy in the objects and its instances. This paper focuses on introducing NeutroGeometry Topological space.