nLab improper filter

Idea

A filter is improper if it is the whole thing … whatever thing the filters are filters in.

Definitions

Definition (improper filter in a lattice or other proset)

If $L$ is a lattice or even a proset, then $L$ is an filter in itself, the improper filter.

Definition (improper filter on a set)

If $S$ is a set, then the improper filter on $S$ is the power set of $S$.

Of course, the improper filter on $S$ is simply the improper filter in the power set of $S$.

Properties

The improper filter contrasts with proper filters (all of the other filters).

The improper filter does not count as a prime filter or an ultrafilter, because it is too simple to be simple, although it may satisfy careless or naïve definitions of those concepts (which must be required to be proper).

Conversely, sometimes one defines ‘filter’ to exclude the improper filter, especially in analysis and topology; however, this makes the set of all filters less nice to work with.