## 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 .

Proof:

- 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.

### Filter prebases

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.