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
.
Proof:
- 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).
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.