Topology is sometimes called "rubber-sheet geometry" because if a shape is drawn on a rubber sheet (like a piece of a balloon), then all of the shapes you can make by stretching or twisting—but never tearing—the sheet are considered to be topologically the same.

Topological properties are based on elastic motions rather than rigid motions like rotations or inversions. Mathematicians are interested in the qualities of figures that remain unchanged even when they are stretched and distorted. The qualities that are unchanged by such transformations are said to be topologically invariant because they do not vary, or change, when stretched.

As an example, the figure shows a triangle, a square, a rough outline of the United States, and a ring. The first three shapes are topologically equivalent; we can stretch and pull the boundary of the square until it becomes a circle or the U.S...