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How does removing points distinguish a circle from a figure-eight curve?

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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.

In topology, removing points is a fundamental method for distinguishing between different geometric figures because connectedness is a topological property that remains unchanged under homeomorphism.12

A circle is a simple closed curve. If you remove a single point from a circle, it remains connected. It only becomes disconnected if two points are removed.12

In contrast, a figure-eight curve is not homeomorphic to a circle because its connectivity changes differently when points are removed. Specifically, if you remove the central crossing point of a figure-eight, the figure becomes disconnected, splitting into two separate components. Additionally, a figure-eight does not share the circle's property of remaining connected after the removal of any single point, as one could remove a point from each loop to alter its structure. These differing responses to the removal of points demonstrate that the two shapes are not topologically equivalent.21