On Topologies Induced by Order Relations on Dual Numbers

Inequalities are often utilized in different areas of mathematics to establish theorems. The presence of inequalities plays a crucial role in the foundations of these branches. In this study, we examine the properties of order relations in the dual number system, inspired by the characteristics of order relations defined on real numbers. With the help of these relations, topologies have been constructed in the space $\mathbb{D}^n$, and the geometric modelings of some elements of these topologies have been carried out.