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.
Definitions
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
.