topological invariant

A topological invariant is a property of a geometric object that remains unchanged even when the object is continuously deformed. This concept is central to topology, a branch of mathematics sometimes called "rubber sheet geometry." In topology, objects are considered equivalent if they can be transformed into one another through bending, twisting, or stretching, without tearing or gluing.

Key topological properties include connectedness and compactness. For example, a simple polygon and a circle are topologically equivalent because one can be continuously deformed into the other. However, a sphere and a torus (doughnut shape) are not, as a torus has a hole that cannot be removed by continuous deformation alone. These invariants help mathematicians classify and understand the fundamental nature of shapes.12