definition of a filter. It is common, though not universal, to require filters on sets to be proper (whatever one's stance on poset filters); we shall again eschew this convention.

Prefilters on a set are proper if and only if they do not contain either.

For every subset of , there is a smallest filter containing . As with prefilters, is said to generate or span ; a base for is the set of all finite intersections of . The set is said to be a filter subbase when (and thus ) is proper.

Proper

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