# closed complex plane

The complex plane^{} $\u2102$, i.e. the set of the complex numbers^{} $z$ satisfying

$$ |

is open but not closed, since it doesn’t contain the accumulation points^{} of all sets of complex numbers, for example of the set $\{1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots}\}$. One can $\u2102$ to the closed complex plane $\u2102\cup \{\mathrm{\infty}\}$ by adding to $\u2102$ the infinite point $\mathrm{\infty}$ which the lacking accumulation points. One settles that $|\mathrm{\infty}|=\mathrm{\infty}$, where the latter $\mathrm{\infty}$ means the real infinity.

The resulting space is the one-point compactification of $\u2102$. The open sets are the open sets in $\u2102$ together with sets containing $\mathrm{\infty}$ whose complement is compact in $\u2102$. Conceptually, one thinks of the additional open sets as those open sets “around $\mathrm{\infty}$”.

The one-point compactification of $\u2102$ is also the complex projective line $\u2102{\mathbb{P}}^{1}$, as well as the Riemann sphere.

Title | closed complex plane |
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Canonical name | ClosedComplexPlane |

Date of creation | 2013-03-22 17:37:48 |

Last modified on | 2013-03-22 17:37:48 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 54E35 |

Classification | msc 30-00 |

Synonym | extended complex plane |

Related topic | RiemannSphere |

Related topic | StereographicProjection |

Related topic | RegularAtInfinity |