# limit points of sequences

In a topological space^{} $X$, a point $x$ is a *limit point ^{} of the sequence* ${x}_{0},{x}_{1},\mathrm{\dots}$ if, for every open set containing $x$, there are finitely many indices such that the corresponding elements of the sequence do not belong to the open set.

A point $x$ is an *accumulation point ^{} of the sequence* ${x}_{0},{x}_{1},\mathrm{\dots}$ if, for every open set containing $x$, there are infinitely many indices such that the corresponding elements of the sequence belong to the open set.

It is worth noting that the set of limit points of a sequence can differ from the set of limit points of the set of elements of the sequence. Likewise the set of accumulation points of a sequence can differ from the set of accumulation points of the set of elements of the sequence.

Reference: L. A. Steen and J. A. Seebach, Jr. “Counterxamples in Topology^{}” Dover Publishing 1970

Title | limit points of sequences |
---|---|

Canonical name | LimitPointsOfSequences |

Date of creation | 2013-03-22 14:38:13 |

Last modified on | 2013-03-22 14:38:13 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 7 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 54A05 |

Defines | limit point of a sequence |

Defines | limit point of the sequence |

Defines | accumulation point of a sequence |

Defines | accumulation point of the sequence |