Definition of a filter
Let be a set.
Traditional definition of a filter on : A collection
of subsets of
is a filter on
iff
; and
; and
.
Alternative definition: A collection of subsets of
is a filter on
iff
; and
.
The two definitions are equivalent.
Definition of 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.