# 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