# examples of filters

• If $X$ is any set and $A\subseteq X$ then $\mathcal{F}=\{F\subseteq X\colon A\subseteq F\}$ 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 $\{F\subseteq X\colon X\smallsetminus F\mbox{is finite }\}$ is a free filter on $X$, called the cofinite filter.

• The filter on $\mathbb{R}$ generated by the filter base $\{(n,\infty)\colon n\in\mathbb{N}\}$ 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 $\{(0,\varepsilon)\colon\varepsilon>0\}$ is a free filter on $\mathbb{R}$ which converges to $0$.

Title examples of filters ExamplesOfFilters 2013-03-22 12:54:31 2013-03-22 12:54:31 Evandar (27) Evandar (27) 8 Evandar (27) Example msc 54A99 msc 03E99 cofinite filter Fréchet filter