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Why are tearing and gluing excluded from continuous deformations in topology?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In topology, tearing and gluing are excluded from continuous deformations because they violate the fundamental requirement of a homeomorphism.

A continuous deformation must be represented by a function that is a one-to-one correspondence, where both the function and its inverse are continuous. Mathematically, continuity ensures that "neighborhoods" are preserved: points that are close together in the original object must remain close together in the transformed object.123

Tearing an object creates a break, which destroys the neighborhood structure by separating points that were previously connected. Conversely, gluing points together collapses distinct points into one, which prevents the mapping from being one-to-one. Because these actions fundamentally alter the topological structure—such as changing the number of components or the connectivity of the space—they are not considered continuous deformations. Instead, topology focuses on properties that remain invariant under stretching, shrinking, and folding, as these transformations preserve the essential topological structure of the object.214