Topological equivalence refers to a concept in mathematics that determines whether two objects or spaces are [[Equivalence|equivalent]] based on their topological properties. In topology, the focus is on studying the properties of spaces that are preserved under continuous transformations, such as stretching, bending, twisting, or [[Deformable|deforming]]. Two objects or spaces are considered topologically equivalent if there exists a continuous transformation (also known as a homeomorphism) between them. This means that one object can be transformed into the other by stretching, bending, and deforming without tearing or gluing any parts. Topological equivalence is determined by examining certain fundamental properties of spaces, such as connectedness, compactness, and continuity. For example, a circle and a square are topologically equivalent because they can be transformed into each other without tearing or gluing. Similarly, a sphere and a cube are also topologically equivalent. The concept of topological equivalence is essential in various areas of mathematics and physics. It helps classify different shapes and spaces into distinct categories based on their underlying topology. It also allows mathematicians to study the properties of spaces regardless of specific geometric details, focusing instead on their essential structural characteristics.