# every net has a universal subnet

Theorem - (Kelley’s theorem) - Let $X$ be a non-empty set. Every net ${({x}_{\alpha})}_{\alpha \in \mathcal{A}}$ in $X$ has a universal^{} subnet (http://planetmath.org/Ultranet). That is, there is a subnet such that for every $E\subseteq X$ either the subnet is eventually in $E$ or eventually in $X-E$.

Proof : Let $\mathcal{F}$ be a section filter for the net ${({x}_{\alpha})}_{\alpha \in \mathcal{A}}$.

Let $\mathcal{D}=\{(\alpha ,U):\alpha \in \mathcal{A},U\in \mathcal{F},{x}_{\alpha}\in U\}$. $\mathcal{D}$ is a directed set^{} under the order relation given by

$$(\alpha ,U)\le (\beta ,V)\u27fa\{\begin{array}{cc}\alpha \le \beta \hfill & \\ V\subseteq U\hfill & \end{array}$$ |

The map $f:\mathcal{D}\u27f6\mathcal{A}$ defined by $f(\alpha ,U):=\alpha $ is order preserving and cofinal. Therefore there is a subnet ${({y}_{(\alpha ,U)})}_{(\alpha ,U)\in \mathcal{D}}$ of ${({x}_{\alpha})}_{\alpha \in \mathcal{A}}$ associated with the map $f$ (that is, ${y}_{(\alpha ,U)}={x}_{\alpha}$).

We now prove that ${({y}_{(\alpha ,U)})}_{(\alpha ,U)\in \mathcal{D}}$ is a net.

Let $E\subseteq X$. We have that ${({y}_{(\alpha ,U)})}_{(\alpha ,U)\in \mathcal{D}}$ is frequently in $E$ or frequently in $X-E$.

Suppose ${({y}_{(\alpha ,U)})}_{(\alpha ,U)\in \mathcal{D}}$ is frequently in $E$.

Let $A\in \mathcal{F}$ and $S(\alpha ):=\{{x}_{\beta}:\alpha \le \beta \}$. We have that $S(\alpha )\in \mathcal{F}$ by definition of section filter.

As $\mathcal{F}$ is a filter, $A\cap S(\alpha )\ne \mathrm{\varnothing}$ and so there exists $\beta $ with $\alpha \le \beta $ such that ${x}_{\beta}\in A$. Hence, $(\beta ,A)\in \mathcal{D}$.

As ${({y}_{(\alpha ,U)})}_{(\alpha ,U)\in \mathcal{D}}$ is frequently in $E$, there exists $(\gamma ,B)\in \mathcal{D}$ with $(\beta ,A)\le (\gamma ,B)$ such that ${y}_{(\gamma ,B)}\in E$.

Also, ${y}_{(\gamma ,B)}$ is in $B$, and therefore, in $A$. So $A\cap E\ne \mathrm{\varnothing}$.

We conclude that $E\cap A\ne \mathrm{\varnothing}$ for every $A\in \mathcal{F}$. Therefore, $\mathcal{F}\cup \{E\}$ a filter in $X$. As $\mathcal{F}$ is a maximal filter we conclude that $E\in \mathcal{F}$, and consequently, $(\gamma ,E)\in \mathcal{D}$.

We can now see that for every $(\delta ,C)$ with $(\gamma ,E)\le (\delta ,C)$, ${y}_{(\delta ,C)}$ is in $C$ and so is in $E$. Therefore, ${({y}_{(\alpha ,U)})}_{(\alpha ,U)\in \mathcal{D}}$ is eventually in $E$.

*Remark:* If ${({y}_{(\alpha ,U)})}_{(\alpha ,U)\in \mathcal{D}}$ is frequently in $X-E$, by an analogous we can conclude that it is eventually in $X-E$.

This proves that ${({y}_{(\alpha ,U)})}_{(\alpha ,U)\in \mathcal{D}}$ is a subnet of ${({x}_{\alpha})}_{\alpha \in \mathcal{A}}$. $\mathrm{\square}$

Title | every net has a universal subnet |
---|---|

Canonical name | EveryNetHasAUniversalSubnet |

Date of creation | 2013-03-22 17:25:16 |

Last modified on | 2013-03-22 17:25:16 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 8 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 54A20 |

Synonym | Kelley’s theorem |

Related topic | Ultranet |