examples of filters

•
If $X$ is any set and $A\subseteq X$ then $\mathcal{F}=\{F\subseteq X:A\subseteq F\}$ is a fixed filter on $X$; $\mathcal{F}$ is an ultrafilter^{} iff $A$ consists of a single point.

•
If $X$ is any infinite set^{}, then $\{F\subseteq X:X\setminus F\text{is finite}\}$ is a free filter on $X$, called the cofinite filter.

•
The filter on $\mathbb{R}$ generated by the filter base $\{(n,\mathrm{\infty}):n\in \mathbb{N}\}$ is called the Fréchet filter on $\mathbb{R}$; it is a free filter which does not converge^{} or have any accumulation points^{}.

•
The filter on $\mathbb{R}$ generated by the filter base $\{(0,\epsilon ):\epsilon >0\}$ is a free filter on $\mathbb{R}$ which converges to $0$.
Title  examples of filters 

Canonical name  ExamplesOfFilters 
Date of creation  20130322 12:54:31 
Last modified on  20130322 12:54:31 
Owner  Evandar (27) 
Last modified by  Evandar (27) 
Numerical id  8 
Author  Evandar (27) 
Entry type  Example 
Classification  msc 54A99 
Classification  msc 03E99 
Defines  cofinite filter 
Defines  Fréchet filter 