Britannica Sources used in this answer

Why can a point inside a circle not be moved outside the circle within a plane without cutting the circle?

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 reason a point inside a circle cannot be moved outside without crossing the boundary is explained by the Jordan curve theorem.

This fundamental theorem states that any simple closed curve—a continuous curve that does not intersect itself—divides the plane into exactly two distinct regions: one inside and one outside. Because the curve serves as the common boundary between these two regions, any continuous path connecting a point in the interior to a point in the exterior must necessarily intersect the curve itself.12

In mathematical terms, moving the point without cutting the circle would require a continuous transformation that preserves the separation of these regions. Since the circle acts as a barrier that defines the topology of the plane, the interior and exterior remain topologically distinct, and a point cannot transition between them without passing through the boundary.1