## Lower and upper limits of a filter base

Two cases, that I have been unable to unify: a filter base over a power set with a set, or over .

## First case: filter bases over a power set

filter base over , with a set.

### Property

It is always the case that .

Proof: Let be an eventual element of , union for all of the sets .

Since is in this union of sets, it is in at least one of them: there exists an such that , that is, such that for all , belongs to .

Now let be any element of . Since and are both in the filter base , their intersection is non-empty and there exists an in this intersection.

Since , we have seen that .

Furthermore, .

Hence .

This being the case for any , we have , that is, .

This being the case for all , we have .

### Property

If and are filter bases over a same , with finer than , then

That is, the lower limits increase with the fineness of the filter base, and the upper limits decrease.

Proof: is finer than , that is, every is a superset of some .

Let be an eventual element of , that is: . Then there is some such that is in every . Since is finer than , is a superset of some . Then is in every , that is, , with . Hence .

This being the case for all , we have .

Let be an eventual element of , that is, . Let be an element of . Since is finer than , is a superset of some . Since , we have , that is, there is an element such that . Since , we also have , and hence . This being the case for any element of , we have .

This being the case for all , we have .

### Property

If two filter bases are equivalent, they have the same upper limits and lower limits.

Proof: If the filter bases are equivalent, they are each finer than the other, hence the lower limit of one is both subset and superset of that of the other, that is, are equal; the same for the upper limits.

### Sequences of sets

#### Definitions and expressions of the upper and lower limits of a sequence of sets

Let be a sequence of sets, all subsets of a set .

The upper and lower limits and of this sequence are classically defined as:

is the set of elements that are in for an infinity of values of ;

is the set of elements that are in for all but a finite number of .

If , then for any , there is a such that , that is, . Conversely, if for any we have , then . Hence:

If , there must be an such that all the values of for which are less than ; then for all , we have , that is, . Conversely, if there is an such that , then all values of such that must be less than , and hence there are a finite number of them; thus . Hence:

#### Fréchet filter and filter base

The Fréchet filter is the filter on the elements of which are the cofinite subsets of . Let be defined as the set of subsets for all . It is easily checked that is a filter base and generates on ; we will call it the Fréchet filter base.

#### Fréchet filter and upper/lower limits of a sequence of sets

Again, let be a sequence of subsets of a set .

We can consider as a mapping ; the Fréchet filter base has a certain image by this mapping, that is a filter base on . An element of corresponds to the element .