The complex plane $\mathbb{C}$ , i.e. the set of the complex numbers $z$ satisfying $$|z| < \infty,$$ is open but not closed, since it doesn't contain the accumulation points of all sets of complex numbers, for example of the set $\{1,\,2,\,3,\,\ldots\}$ . One can supplement $\mathbb{C}$ to the closed complex plane $\mathbb{C}\cup\{\infty\}$ by adding to $\mathbb{C}$ the infinite point $\infty$ which represents the lacking accumulation points. One settles that $|\infty| = \infty$ , where the latter $\infty$ means the real infinity.

The resulting space is the one-point compactification of $\mathbb{C}$ . The open sets are the open sets in $\mathbb{C}$ together with sets containing $\infty$ whose complement is compact in $\mathbb{C}$ . Conceptually, one thinks of the additional open sets as those open sets ``around $\infty$ ''.

The one-point compactification of $\mathbb{C}$ is also the complex projective line $\mathbb{CP}^1$ , as well as the Riemann sphere.