PlanetMath: examples of filters

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examples of filters

(Example)

If is any set and $A\subseteq X$ then $\mathcal{F} = \{ F\subseteq X\colon A\subseteq F\}$ is a fixed filter on ; $\mathcal{F}$ is an ultrafilter iff consists of a single point.
If is any infinite set, then $\{ F\subseteq X\colon X\smallsetminus F$ is finite $\}$ is a free filter on , 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.

"examples of filters" is owned by Evandar.

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Also defines:

cofinite filter, Fréchet filter

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Cross-references: accumulation points, converge, filter base, generated by, filter, free filter, infinite set, point, iff, ultrafilter, fixed filter
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This is version 5 of examples of filters, born on 2002-08-02, modified 2004-03-31.
Object id is 3259, canonical name is ExampleOfFilters.
Accessed 4653 times total.

Classification:

AMS MSC:	54A99 (General topology :: Generalities :: Miscellaneous)
	03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)

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