ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-07-31

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The subsets of $X$ containing a fixed point $x$ form the principal ultrafilter at $x$

Example

Let $x \in X$ . The family

$U_{x} : = {A \subseteq X : x \in A}$

is the principal ultrafilter at $x$ . It is the filter generated by the singleton ${x}$ .

Facts & Assumptions

Given: A set $X$ , a point $x \in X$ , and the family $U_{x}$ displayed above.

[L1]

If $\emptyset \neq C \subseteq X$ , then $F_{C} = {A \subseteq X : C \subseteq A}$ is the filter generated by ${C}$ , and it is an ultrafilter exactly when $C$ is a singleton (For every nonempty $C \subseteq X$ , the supersets of $C$ form the filter generated by ${C}$ , and this filter is an ultrafilter exactly when $C$ is a singleton).

[F1]

A principal ultrafilter on $X$ is an ultrafilter of the form ${A \subseteq X : y \in A}$ for some $y \in X$ (Ultrafilter).

[L2]

A filter is an ultrafilter exactly when it contains exactly one of $A$ and $X ∖ A$ for every $A \subseteq X$ (Characterisation of ultrafilters: every set or its complement).

Verification

technique · direct

1.1

The set ${x}$ is a nonempty singleton, and ${x} \subseteq A$ holds exactly when $x \in A$ .

given

1.2

For every $A \subseteq X$ , exactly one of $x \in A$ and $x \in X ∖ A$ holds.

given

2.1

Applying [L1] to $C = {x}$ shows that $U_{x}$ is the filter generated by ${{x}}$ and is an ultrafilter.

step 1.1L1

3.1

Equivalently, step 1.2 verifies the complementary-pair condition of [L2] directly.

step 1.2step 2.1L2

4.1

By [F1], this ultrafilter is principal, and the displayed formula identifies it as the principal ultrafilter at $x$ .

step 2.1F1∎

Depends on

Used by

Dependency tree · two levels

9 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

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

The subsets of XX containing a fixed point xx form the principal ultrafilter at xx