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How can a point be moved from inside a circle to outside it 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 premise that a point must pass through a circle to move from its inside to its outside is a property specific to two-dimensional space, as defined by the Jordan curve theorem. In a two-dimensional plane, a simple closed curve acts as a boundary that separates the space into two distinct regions; any path connecting a point in the interior to a point in the exterior must intersect the curve.12

However, this constraint does not apply in three-dimensional space. Because a circle is a one-dimensional object embedded in a higher-dimensional environment, it does not "divide" three-dimensional space into separate regions. To move a point from the inside of a circle to the outside in three-dimensional space, you simply move the point along a path that travels "around" or "over" the circle through the third dimension. By lifting the point out of the plane containing the circle, you can bypass the boundary entirely and return to the plane on the opposite side of the curve without ever touching it.