Topological invariants are properties of a geometric object that remain unchanged under continuous deformations, such as stretching or bending, but not tearing or gluing. These properties help classify shapes and spaces in the field of topology, a branch of mathematics that studies the properties of space that are preserved under continuous transformations. Common examples of topological invariants include the Euler characteristic, which relates to the number of vertices, edges, and faces in a polyhedron, and the homotopy groups, which describe the different ways a space can be continuously transformed. These invariants are essential for understanding the fundamental structure of various mathematical and physical systems.