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How can different Euler characteristics prove that two surfaces are not topologically equivalent?

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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, the Euler characteristic is a topological invariant, meaning it remains unchanged when a surface is stretched or deformed without tearing. Because this value is intrinsic to the shape's structure, two surfaces that are homeomorphic—or topologically equivalent—must share the same Euler characteristic.123

The characteristic is calculated using the formula $χ = V - E + F$ , where $V$ , $E$ , and $F$ represent the number of vertices, edges, and faces of a network dividing the surface. For a surface with $g$ holes, such as a $g$ -holed torus, the characteristic is generalized as $χ = 2 - 2 g$ .13

Consequently, if two surfaces possess different Euler characteristics, they cannot be deformed into one another through continuous transformation. This property serves as a fundamental tool in algebraic topology to distinguish between non-equivalent topological spaces.3