Why must a Klein bottle intersect itself when represented in three-dimensional space?

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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.

The Klein bottle is a topological space that cannot be constructed in three-dimensional Euclidean space without intersecting itself because it is a non-orientable surface.1

In topology, the Klein bottle is formed by identifying the ends of a cylinder in a way that reverses the orientation, a process that requires a fourth dimension to complete without the surface passing through itself. When forced into three-dimensional space, the surface must necessarily pass through its own wall to connect the ends as required by its definition, resulting in the self-intersection commonly seen in physical models. Unlike a sphere or a torus, the Klein bottle is one-sided, similar to a Möbius strip, and lacks a distinct "inside" or "outside".1