PlanetMath: comparison of filters

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comparison of filters

(Definition)

Let $\mathbb{F}_1$ and $\mathbb{F}_2$ be two filters on the same set. The following terminology is commonly used to describe the relation of $\mathbb{F}_1$ to $\mathbb{F}_2$ :

$\mathbb{F}_2$ is said to be finer than $\mathbb{F}_1$ if $\mathbb{F}_1 \subseteq \mathbb{F}_2$ .

$\mathbb{F}_2$ is said to be coarser than $\mathbb{F}_1$ if $\mathbb{F}_1 \supseteq \mathbb{F}_2$ .

$\mathbb{F}_2$ is said to be strictly finer than $\mathbb{F}_1$ if $\mathbb{F}_1 \subset \mathbb{F}_2$ .

$\mathbb{F}_2$ is said to be strictly coarser than $\mathbb{F}_1$ if $\mathbb{F}_1 \supset \mathbb{F}_2$ .

$\mathbb{F}_1$ and $\mathbb{F}_2$ are said to be comparable if either $\mathbb{F}_1 \subseteq \mathbb{F}_2$ or $\mathbb{F}_1 \supseteq \mathbb{F}_2$ .

"comparison of filters" is owned by rspuzio.

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Also defines:

finer, coarser, strictly finer, strictly coarser, comparable

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Cross-references: relation, filters
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This is version 2 of comparison of filters, born on 2004-10-06, modified 2004-10-07.
Object id is 6303, canonical name is ComparisonOfFilters.
Accessed 5915 times total.

Classification:

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

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