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.