comparison of filters

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 $\mathbb{F}_{1}\supseteq\mathbb{F}_{2}$ .

Title	comparison of filters
Canonical name	ComparisonOfFilters
Date of creation	2013-03-22 14:41:38
Last modified on	2013-03-22 14:41:38
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	5
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 54A99
Classification	msc 03E99
Defines	finer
Defines	coarser
Defines	strictly finer
Defines	strictly coarser
Defines	comparable