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# closed complex plane

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.

## Mathematics Subject Classification

54E35*no label found*30-00

*no label found*

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