I am beginning to study Filters and Ultrafilters but I am not clear about two things.
(i) Why a set $X$ must be no empty in Filters and Ultrafilters? I think it is due to limitations with the empty set.
(ii) Where does the name "filter" come from?
Thanks.
(i)
A filter $\mathcal F$ on a set $X$ is not empty by definition, since one of the conditions is that $\mathcal F\subseteq\mathcal P(X)$ must contain $X\in \mathcal F$ itself.
If you instead mean why $\varnothing\notin \mathcal F$ must be true, this depends on personal taste. Some authors exclude it, since a filter containing $\varnothing$ is automatically equal to $\mathcal P(X)$ itself (filters are closed under supersets) and thus is "uninteresting" for most applications, or worse, it might be the only counterexample to some theorems about filters. These authors call the filter that contains $\varnothing$ the trivial filter, and call all other filters proper filters. Some other authors put $\varnothing\notin\mathcal F$ in the definition of a filter.
It's comparable to how $1$ is not considered a prime number, and other such pathological edge cases; leaving them out will spare you the work of mentioning every time that you only consider nontrivial cases.
As an example, "every filter can be extended to an ultrafilter" is not true for the trivial filter, but "every proper filter can be extended to an ultrafilter" is true (as a consequence of Zorn's lemma).
An ultrafilter is by definition proper, so it does not contain $\varnothing$.
(ii)
There are some answers to be found on this page.
The word filter stems from the French filtre, probably introduced by Henri Cartan, although that text does not seem to mention an etymology. It could be that I can't find it, since my French is rather limited.
See also answers over here.
