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How does topology study shapes that are stretched or folded without tearing?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

Topology, often called "rubber sheet geometry," studies shapes by focusing on properties that remain unchanged when an object is continuously deformed—such as bending, twisting, stretching, or shrinking—without tearing or gluing.12

Unlike traditional geometry, which often measures specific quantities like lengths or angles, topology examines qualitative features. Two objects are considered topologically equivalent if one can be transformed into the other through a homeomorphism, which is a one-to-one mapping that is continuous in both directions.134

Properties preserved under these deformations are known as topological invariants, such as connectedness or compactness. Additionally, the concept of isotopy considers how these deformations occur within a specific "ambient space." For instance, an object that cannot be deformed into another in two-dimensional space might be able to do so in three or four dimensions, as the surrounding space dictates the possibilities for movement.35