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Under the ultrafilter lemma, every ultrafilter algebra determines a compact Hausdorff topology

Statement

Assume UL/BPI. For every ultrafilter algebra $ξ : β X \to X$ , the topology induced by $ξ$ is compact and Hausdorff.

Facts & Assumptions

Given: UL/BPI and an ultrafilter algebra $ξ : β X \to X$ .

[L1]

The open-set family induced by an ultrafilter algebra is a topology (The open-set family induced by an ultrafilter algebra is a topology).

[L2]

Under UL/BPI, every ultrafilter has exactly one limit in the induced topology, namely its algebra value (Under the ultrafilter lemma, an ultrafilter algebra maps each ultrafilter to its unique limit).

[L3]

Under the ultrafilter lemma, a space is compact if and only if every ultrafilter on it converges (Assuming the ultrafilter lemma, compactness is equivalent to every net having a cluster point, every net having a convergent subnet, every filter having a cluster point, and every ultrafilter converging).

[L4]

The ultrafilter extension principle says that every filter on a set is contained in an ultrafilter on that set (The ultrafilter extension principle (UL/BPI)).

Proof

technique · direct

1.1

Equip $X$ with the topology supplied by [L1].

1.2

By [L2], every ultrafilter on $X$ converges, and its limit is unique.

2.1

The equivalence in [L3] applied to step 1.2 proves that the induced topology is compact.

step 1.2L3

2.2

If distinct $x, y$ had no disjoint neighbourhoods, the union of their two neighbourhood filters would have the finite-intersection property. By [L4] it extends to an ultrafilter converging to both $x$ and $y$ , contradicting uniqueness in step 1.2. Hence the topology is Hausdorff, with the empty and singleton cases vacuous.

step 1.2L4choose

3.1

Steps 2.1 and 2.2 prove that the induced topology is compact Hausdorff.

step 2.1step 2.2∎

Depends on

Used by

Under the ultrafilter lemma, compact Hausdorff spaces and ultrafilter algebras are recovered by the two limit constructions Theorem

Dependency tree · two levels

23 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

J. Goubault-Larrecq, Algebras of filter-related monads: I. Ultrafilters and Manes' theorem (standard reference, not scraped)