*Remark: To forestall empty set difficulties, whenever I talk about arbitrary sets in this post they will be non-empty.*

We continue our exploration of ultrafilters from the previous post. Recall that a (proper) **filter** on a poset is a non-empty subset such that

- For every , there is some such that .
- For every , if then .
- is not the whole set .

If has finite infima (meets), then the first condition, given the second, can be replaced with the condition that if then . (This holds in particular if is the poset structure on a Boolean ring, in which case .) A filter is an **ultrafilter** if in addition it is maximal under inclusion among (proper) filters. For Boolean rings, an equivalent condition is that for every either or , but not both. Recall that this condition tells us that ultrafilters are precisely complements of maximal ideals, and can be identified with morphisms in . If for some set , then we will sometimes call an ultrafilter on an ultrafilter on (for example, this is what people usually mean by “an ultrafilter on “).

Today we will meander towards an ultrafilter point of view on topology. This point of view provides, among other things, a short, elegant proof of Tychonoff’s theorem.