topological equivalence

Topological equivalence, also known as homeomorphism, describes the relationship between two geometric figures that can be transformed into one another through continuous deformation. This means they can be bent, stretched, or twisted without tearing or gluing.12

Key characteristics of topologically equivalent figures include:
* A one-to-one mapping between their points.
* This mapping and its inverse must be continuous.2

Properties preserved under such transformations are called topological properties. Examples include connectedness and compactness. For instance, a simple polygon and a circle are topologically equivalent because a rubber band shaped like the polygon can be stretched to fit the circle. However, a sphere and a torus (doughnut shape) are not topologically equivalent, as loops on a sphere can always be shrunk to a point, while some loops on a torus cannot.23