examples of filters

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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.
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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.
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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.
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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 $0$ .