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The open-set family induced by an ultrafilter algebra
Definition
Let be an algebra (Algebra and algebra homomorphism for a monad) for the ultrafilter monad, that is, for the endofunctor with its principal unit and flattening multiplication, which form a monad on by The ultrafilter endofunctor with principal unit and flattening multiplication is a monad. A subset is -open, or induced-open, when
for every ultrafilter on . Write for the family of all -open subsets.
The topology induced by is . The fact that this family satisfies the topology axioms is proved in The open-set family induced by an ultrafilter algebra is a topology ↗.
Depends on
- Algebra and algebra homomorphism for a monad
- The ultrafilter endofunctor with principal unit and flattening multiplication
- The ultrafilter endofunctor with principal unit and flattening multiplication is a monad
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
Used by
- The open-set family induced by an ultrafilter algebra is a topology Lemma
- Under the ultrafilter lemma, an ultrafilter algebra maps each ultrafilter to its unique limit Lemma
- Under the ultrafilter lemma, closure in an ultrafilter-algebra topology is the image of ultrafilters containing the set Lemma
- Under the ultrafilter lemma, compact Hausdorff spaces and ultrafilter algebras are recovered by the two limit constructions Theorem
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
- J. Goubault-Larrecq, Algebras of filter-related monads: I. Ultrafilters and Manes' theorem (standard reference, not scraped)