How did Bernhard Riemann use topological ideas in complex number theory?

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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.

Bernhard Riemann utilized topological concepts to advance complex analysis by introducing the Riemann surface. In his 1851 doctoral thesis, he addressed the challenge of representing equations involving two complex variables, which define a real surface spread over the complex plane.1

To classify these surfaces, Riemann employed a topological invariant now known as the genus. This value is determined by the maximum number of non-intersecting closed curves that can be drawn on the surface without dividing it into separate pieces. This work is recognized as one of the earliest significant applications of topology in mathematics, shifting the focus toward conceptual understanding rather than purely algebraic calculation.1