## 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.