We would like a way to avoid modifying the domain every time we choose a Möbius Transformation, as this becomes messy especially if we want to compose Möbius transformations. To do this, we introduce the One Point Compactification of the complex plane, which we denote $\hat{\mathbb{C}}$. What Topology do we put on this space?
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The Open sets of $\hat{\mathbb{C}}=\mathbb{C}\coprod \{\mathrm{\infty }\}$ consists of all open sets of $\mathbb{C}$ in addition to complements of compact sets $K\subseteq \mathbb{C}$.

Concerning arithmetic operations, we now consder $\frac{1}{0}=\mathrm{\infty }$, and $\frac{1}{\mathrm{\infty }}=0$. This allows us to consider every Möbius transformation:
$f:\hat{\mathbb{C}}\to \hat{\mathbb{C}}$, so we don't need to change the domain every time.