Britannica Sources used in this answer

Which kinds of shape changes are allowed in topology?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

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, often called "rubber sheet geometry," objects are considered equivalent if they can be transformed into one another through continuous deformations. These allowed shape changes include:1

Bending1
Twisting1
Stretching12
Shrinking12
Folding2

These transformations must be performed without tearing the object apart or gluing together parts that were previously separate.12

Formally, this relationship is defined by a homeomorphism, which is a one-to-one mapping between two objects that is continuous in both directions. Because topology focuses on these qualitative properties rather than rigid measurements like length or angle, objects that can be continuously deformed into one another are considered topologically equivalent.341

Additionally, the ability to perform these deformations may depend on the ambient space in which the objects are embedded; if a transformation is possible within a specific space, the objects are said to be isotopic.5