Topology invariants are properties of a topological space that remain unchanged under continuous transformations, such as stretching or bending, but not tearing or gluing. These invariants help classify spaces into different types, allowing mathematicians to understand their fundamental structure. Common examples include the number of holes in a shape, which is captured by the concept of homology, and the Euler characteristic, which relates to the number of vertices, edges, and faces in a polyhedron. Invariants are crucial in various fields, including algebraic topology and differential geometry, as they provide essential information about the space's shape and connectivity. By studying these properties, mathematicians can determine whether two seemingly different spaces are topologically equivalent, meaning they can be transformed into one another through continuous deformations.