Category:Filters on Sets
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This category contains results about Filters on Sets.
Definitions specific to this category can be found in Definitions/Filters on Sets.
Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
A set $\FF \subset \powerset S$ is a filter on $S$ if and only if $\FF$ satisfies the filter on set axioms:
\((\text F 1)\) | $:$ | \(\ds S \in \FF \) | |||||||
\((\text F 2)\) | $:$ | \(\ds \O \notin \FF \) | |||||||
\((\text F 3)\) | $:$ | \(\ds U, V \in \FF \implies U \cap V \in \FF \) | |||||||
\((\text F 4)\) | $:$ | \(\ds \forall U \in \FF: U \subseteq V \subseteq S \implies V \in \FF \) |
Subcategories
This category has only the following subcategory.
E
- Examples of Filters on Sets (2 P)