# accumulation points and convergent subnets

###### Proposition.

Let $X$ be a topological space^{} and ${\mathrm{(}{x}_{\alpha}\mathrm{)}}_{\alpha \mathrm{\in}A}$ a net in $X$. A point $x\mathrm{\in}X$
is an accumulation point^{} of $\mathrm{(}{x}_{\alpha}\mathrm{)}$ if and only if some subnet of $\mathrm{(}{x}_{\alpha}\mathrm{)}$ converges^{} to $x$.

###### Proof.

Suppose first that ${({x}_{{\alpha}_{\beta}})}_{\beta \in B}$ is a subnet of $({x}_{\alpha})$ converging to $x$. Given an open subset $U$ of $X$ containing $x$ and $\alpha \in A$, we may select ${\beta}_{1}\in B$ such that ${x}_{{\alpha}_{\beta}}\in U$ for $\beta \ge {\beta}_{1}$, as well as ${\beta}_{2}\in B$ such that ${\alpha}_{\beta}\ge \alpha $ for $\beta \ge {\beta}_{2}$. Finally, because $B$ is directed, there exists $\beta \in B$ such that $\beta \ge {\beta}_{1}$ and $\beta \ge {\beta}_{2}$; we then have ${\alpha}_{\beta}\ge \alpha $ and ${x}_{{\alpha}_{\beta}}\in U$, so that $({x}_{\alpha})$ is frequently in $U$, whence $x$ is an accumulation point of $({x}_{\alpha})$. Conversely, suppose that $x$ is an accumulation point of $({x}_{\alpha})$, let $N$ be the set of open neighborhoods of $x$ in $X$, directed by reverse inclusion, and let $B=A\times N$, directed in the natural way. For each pair $(\gamma ,U)\in B$, select ${\alpha}_{(\gamma ,U)}\in B$ such that $\alpha \ge \gamma $ and ${x}_{{\alpha}_{(\gamma ,U)}}\in U$; ${({x}_{{\alpha}_{(\gamma ,U)}})}_{(\gamma ,U)\in B}$ is then a subnet of $({x}_{\alpha})$ that converges to $x$, for given $U\in N$ and $\gamma \in A$, if $({\gamma}^{\prime},{U}^{\prime})\ge (\gamma ,U)$, then ${\alpha}_{({\gamma}^{\prime},U)}\ge {\gamma}^{\prime}\ge \gamma $ and ${x}_{{\alpha}_{({\gamma}^{\prime},{U}^{\prime})}}\in {U}^{\prime}\subseteq U$. ∎

Title | accumulation points and convergent subnets |
---|---|

Canonical name | AccumulationPointsAndConvergentSubnets |

Date of creation | 2013-03-22 18:37:40 |

Last modified on | 2013-03-22 18:37:40 |

Owner | azdbacks4234 (14155) |

Last modified by | azdbacks4234 (14155) |

Numerical id | 9 |

Author | azdbacks4234 (14155) |

Entry type | Theorem |

Classification | msc 54A20 |

Related topic | Net |

Related topic | Neighborhood^{} |

Related topic | DirectedSet |

Related topic | CompactnessAndConvergentSubnets |