examples of filters

• If $X$ is any set and $A\subseteq X$ then $ℱ=\{F\subseteq X:A\subseteq F\}$ is a fixed filter on $X$; $ℱ$ 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 $ℝ$ generated by the filter base $\{(n,\mathrm{\infty }):n\in \mathbb{N}\}$ is called the Fréchet filter on $ℝ$; it is a free filter which does not converge or have any accumulation points.

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