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
If and are filters on a same set , then is finer than iff .
Filter bases and filter prebases
A collection of sets is a filter base iff:
- is non-empty;
No enclosing set is specified. Indeed, a filter base generates a filter on any superset of all sets in (that is, on any ) by:
Obviously, the filter that is generated is a superset of the filter base.
Equivalent filter bases
If and are any two filter bases and a common enclosing set, let and be the respective filters generated on .
Then is finer than iff every element of is superset of some element of .
- Assume that is finer than . Let be an element of . Then is also an element of , and hence of . Since the latter is generated by , is a superset of some element of . Hence every element of is a superset of some element of .
- Assume that every element of is a superset of some element of . Let be an element of . Then it is a superset of some element of . This in turn is a superset of some element of . We have , hence with , hence . Hence , that is, is finer than .
It follows that and will generate the same filter on iff both every element of is superset of some element of and every element of is superset of some element of . Then and are said to be equivalent filter bases (a property that does not depend on the choice of the common enclosing set).
It is trivial to check that equivalence is an equivalence relation among filter bases.
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.