Definitions concerning filters
Definition of a filter
Let be a set.
Traditional definition of a filter on : A collection of subsets of is a filter on iff:
- is non-empty (equivalently in context: );
It follows from the definition that there can be no filters on the empty set.
If we waive the second rule and allow among elements of a filter, we obtain a hyperfilter on . The only hyperfilter on that is not a filter is , since if is an element, all supersets of must also belong.
Alternative equivalent definition
A collection of subsets of is a filter on iff is non-empty; and ; and .
Dispensing with the specification of the enclosing set
One can dispense with the specification of the enclosing set , retrieving it as . A filter is then a collection of sets such that:
- is non-empty;
Fineness of filters
Definition: If and are filters on a same set , then is finer than iff .
Filter bases and filter prebases
Definition: A collection of sets is a filter base iff:
- is non-empty;
No enclosing set is specified.
It is trivial to check that a filter is always a filter base.
Filter generated by a filter base
If is a filter base and a superset of all the elements of , let be the set of all subsets of that are superset of some element of :
It is trivial to check that is then a filter on , and is a superset of .
By definition, is the filter generated on by .
Fineness of filter bases
Definition: If and are any two filter bases, then is finer than iff every element of is a superset of some element of .
Property: If and are filters on set and and filter bases that generate respectively and , then is finer than iff is finer than .
- Assume that is finer than . Let be an element of . It is then also an element of . Since , we have , and hence is a superset of some element of . This is true for all , hence is finer than .
- Assume that is finer than . Let be an element of . Then is superset of some element of (since generates ), which itself (since is finer than ) is a superset of some element of , and thus is an element of . This is true for all , hence is finer than .
Equivalent filter bases
Definition: Two filter bases are equivalent if each is finer than the other.
Property: Two filter bases generate the same filter on a common enclosing set iff they are equivalent.
Proof: The generated filters are equal, that is, each finer than the other, iff their generating filter bases are each finer than the other, that is, equivalent.
Property: Equivalence is an equivalence relation among filter bases (trivial).
A nonempty collection of sets is a filter prebase iff all finite intersections of elements of are nonempty: for any finite and nonempty , we have .
If is a filter prebase, then together with all the finite intersections of its elements it forms a filter base.
The image of a filter
Let and be sets, a filter on and .
Traditional definition: The filter on image of by is .
Alternative definition: The filter on image of by is .
Again, the two definitions are equivalent.