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Why is the surface of a sphere not topologically equivalent to a torus?

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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, two objects are considered topologically equivalent if they are homeomorphic, meaning there exists a continuous, one-to-one mapping between them that has a continuous inverse.1

The surface of a sphere and a torus are not topologically equivalent because they possess different fundamental topological properties, specifically their genus. A sphere has a genus of 0, while a torus has a genus of 1. Because a homeomorphism must preserve the essential "neighborhood" structure of a space, it cannot transform an object with a hole (the torus) into an object without one (the sphere) without tearing or gluing, which violates the requirement for a continuous transformation.1

Essentially, the torus contains a non-contractible loop that cannot be shrunk to a point, whereas every loop on the surface of a sphere can be continuously contracted to a single point. Since these structural features are preserved under homeomorphism, the two surfaces remain distinct in the eyes of topology.1