Definition 4

Tag 0B49, permanent. Cite it as [Bourbaki, Tag 0B49].

General Topology, chapter I, § 6. Filters

An ultrafilter on a set $X$ is a filter $\mathfrak{F}$ such that there is no filter on $X$ which is strictly finer than $\mathfrak{F}$ (in other words, a maximal element in the ordered set of all filters on $X$ ).

Since the ordered set of all filters on $X$ is inductive (no. 2, Proposition 1, Corollary 3), Zorn's lemma (Set Theory, R, § 6, no. 10) shows that:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

§ 6, no. 10
no. 2, Proposition 1
Corollary 3