# frequently in

Recall that a net is a function $x$ from a directed set^{} $D$ to a set $X$. The value of $x$ at $i\in D$ is usually denoted by ${x}_{i}$. Let $A$ be a subset of $X$. We say that a net $x$ is *frequently in* $A$ if for every $i\in D$, there is a $j\in D$ such that $i\le j$ and ${x}_{j}\in A$.

Suppose a net $x$ is frequently in $A\subseteq X$. Let $E:=\{j\in D\mid {x}_{j}\in A\}$. Then $E$ is a cofinal subset of $D$, for if $i\in D$, then by definition of $A$, there is $i\le j\in D$ such that ${x}_{j}\in A$, and therefore $j\in E$.

The notion of “frequently in” is related to the notion of “eventually in” in the following sense: a net $x$ is eventually in a set $A\subseteq X$ iff it is not frequently in ${A}^{\mathrm{\complement}}$, its complement^{}. Suppose $x$ is eventually in $A$. There is $j\in D$ such that ${x}_{k}\in A$ for all $k\ge j$, or equivalently, ${x}_{k}\in {A}^{\mathrm{\complement}}$ for no $k\ge j$. The converse^{} is can be argued by tracing the previous statements backwards.

In a topological space^{} $X$, a point $a\in X$ is said to be a *cluster point ^{} of a net* $x$ (or, occasionally, $x$

*clusters at*$a$) if $x$ is frequently in every neighborhood

^{}of $a$. In this general definition, a limit point is always a cluster point. But a cluster point need not be a limit point. As an example, take the sequence $0,2,0,4,0,6,0,8,\mathrm{\dots},0,2n,0,\mathrm{\dots}$ has $0$ as a cluster point. But clearly $0$ is not a limit point, as the sequence diverges in $\mathbb{R}$.

Title | frequently in |
---|---|

Canonical name | FrequentlyIn |

Date of creation | 2013-03-22 17:14:23 |

Last modified on | 2013-03-22 17:14:23 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 6 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 03E04 |

Synonym | clusters at |

Defines | cluster point of a net |