# types of limit points

Let $X$ be a topological space^{} and $A\subset X$ be a subset.

A point $x\in X$ is an *$\omega $-accumulation point ^{} of $A$* if every open set in $X$ that contains $x$ also contains infinitely many points of $A$.

A point $x\in X$ is a *condensation point of $A$* if every open set in $X$ that contains $x$ also contains uncountably many points of $A$.

If $X$ is in addition a metric space, then a *cluster point* of a sequence $\{{x}_{n}\}$ is a point $x\in X$ such that every $\u03f5>0$, there are infinitely many point ${x}_{n}$ such that $$.

These are all clearly examples of limit points^{}.

Title | types of limit points |
---|---|

Canonical name | TypesOfLimitPoints |

Date of creation | 2013-03-22 14:37:50 |

Last modified on | 2013-03-22 14:37:50 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 54A99 |

Defines | $\omega $-accumulation points |

Defines | condensation points |

Defines | cluster points |