CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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If the exclusion of $\emptyset$ is dropped, $P (X)$ becomes the unique maximal improper filter

Statement refuted

Dropping the properness axiom $\emptyset \notin F$ leaves the maximal filters, and therefore the notion of ultrafilter, unchanged.

Call a family a weak filter here if it satisfies the other filter axioms but may contain $\emptyset$ . Then $P (X)$ is the unique improper weak filter on $X$ and the greatest weak filter under inclusion. Consequently it is the unique maximal weak filter, so maximality becomes degenerate if properness is not retained.

Facts & Assumptions

Given: A set $X$ and the enlarged convention in which a weak filter is a family $G \subseteq P (X)$ that contains $X$ , is closed under pairwise intersection, and is closed upward in $X$ , without requiring $\emptyset \notin G$ .

[F1]

Under this library's convention a filter must also omit $\emptyset$ . The competing convention admits exactly the improper object $P (X)$ (Filter on a set).

[F2]

An ultrafilter is maximal for inclusion among the proper filters on $X$ (Ultrafilter).

Counterexample

technique · direct

1.1given

The family $P (X)$ is a weak filter: it contains $X$ and $\emptyset$ , and it is closed under intersections and under taking supersets inside $X$ .

1.2given

If a weak filter $G$ contains $\emptyset$ , then for every $A \subseteq X$ the inclusions $\emptyset \subseteq A \subseteq X$ and upward closure give $A \in G$ . Hence $G = P (X)$ .

2.1step 1.1step 1.2

Thus $P (X)$ is the unique improper weak filter. Since every weak filter is a subfamily of $P (X)$ , it is also the greatest and therefore the unique maximal weak filter.

3.1step 2.1F1F2∎

If maximality were taken in the enlarged class, no proper filter could be maximal because it would be strictly contained in $P (X)$ . This differs from [F2] and refutes the claim that dropping properness leaves ultrafilters unchanged.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Filter (set theory) (Wikipedia) (standard reference, not scraped)
Ultrafilter (Wikipedia) (standard reference, not scraped)

If the exclusion of ∅∅ is dropped, P(X)P(X) becomes the unique maximal improper filter