Topological 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 Euler characteristic, and the homotopy groups that describe how spaces can be continuously deformed. Invariants play a crucial role in various fields, including algebraic topology and differential geometry. They provide essential tools for distinguishing between different topological spaces and understanding their relationships. By studying these invariants, mathematicians can gain insights into complex structures and their properties.