Britannica Sources used in this answer

Why does the surrounding space matter when determining whether two objects are isotopic?

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, the surrounding space—or ambient space—is critical because it defines the constraints under which an object can be continuously deformed. Two objects are considered isotopic if one can be transformed into the other through a continuous deformation within that specific ambient space without the object passing through itself or requiring "surgery" like cutting.1

The ambient space matters because it determines the available "room" for movement. For example, an isolated point inside a circle cannot be moved outside that circle in a two-dimensional plane without cutting the circle. However, in a three-dimensional ambient space, the point can be lifted out of the plane and reinserted on the other side, making the two configurations isotopic in three dimensions but not in two.1

Similarly, a knot that is impossible to untie in three-dimensional space may be easily untied if an additional dimension is available, as the extra space allows the curve to bypass its own segments. Thus, isotopy provides a measure of extrinsic topological equivalence, where the embedding space dictates the possibility of transformation.12