Remark

Tag 0B4E, permanent. Cite it as [Bourbaki, Tag 0B4E].

General Topology, chapter I, § 6. Filters

If $X$ contains at least two elements, there are at least two distinct ultrafilters on $X$ , and therefore the ordered set of filters on $X$ has no greatest element.

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

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Cites

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