# 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 $\mathbb{C}$ to the closed complex plane $\mathbb{C}\cup\{\infty\}$ by adding to $\mathbb{C}$ the infinite point $\infty$ which 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.

Title closed complex plane ClosedComplexPlane 2013-03-22 17:37:48 2013-03-22 17:37:48 pahio (2872) pahio (2872) 5 pahio (2872) Definition msc 54E35 msc 30-00 extended complex plane RiemannSphere StereographicProjection RegularAtInfinity