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Ultrafilters on $$\omega$$ and atoms in the lattice of uniformities. II. (English) Zbl 0657.54023
The authors continue their investigations on the interaction between ultrafilters and uniformities (on a countable set) [see part I reviewed above]. Under CH they study proximally discrete atoms in the lattice of uniformities on $$\omega$$. Various ultrafilters on $$\omega$$ are constructed so that atoms in the lattice of uniformities refining the corresponding ultrafilter-uniformities have special properties. Methods from finite combinatorics are used. The following results are typical. Let $$1\leq s<\omega$$ and let $${\mathcal G}$$ be an ultrafilter on $$\omega$$. Then there is an ultrafilter $${\mathcal F}>{\mathcal G}$$ on $$\omega$$ such that there are precisely s distinct atoms refining $${\mathcal U}_{{\mathcal F}}$$. All these atoms are zero-dimensional. (Here $$>$$ denotes the Rudin-Keisler order.) Let $${\mathcal G}$$ be an ultrafilter on $$\omega$$. Then there exists an ultrafilter $${\mathcal F}>{\mathcal G}$$ on $$\omega$$ such that there are $$2^ c$$ distinct atoms refining $${\mathcal U}_{{\mathcal F}}$$. There exists a uniformity N, generated by an ultrafilter $${\mathcal F}$$ and a partition $${\mathcal R}$$ into finite sets on $$\omega$$, such that (i) $${\mathcal N}$$ is an atom in the lattice of zero-dimensional uniformities on $$\omega$$, (ii) $${\mathcal N}$$ is not an atom in the lattice of all uniformities on $$\omega$$ ; in fact, there are at least $$2^{\omega}$$ pairwise uniformly non- homeomorphic atoms finer than $${\mathcal N}$$, and these atoms are non-zero- dimensional.
All these non-zero-dimensional atoms have the same distal modification, viz. $${\mathcal N}$$. Thus d$${\mathcal A}_ 1\wedge d{\mathcal A}_ 2\neq d({\mathcal A}_ 1\wedge {\mathcal A}_ 2)$$ for any couple $${\mathcal A}_ 1,{\mathcal A}_ 2$$ of these atoms. (For any uniformity $${\mathcal U}$$, $$d{\mathcal U}$$ denotes the distal modification of $${\mathcal U}$$, i.e. the uniformity whose basis consists of all covers of finite order in $${\mathcal U}.)$$
Reviewer: H.-P.Künzi

##### MSC:
 54E15 Uniform structures and generalizations 03E50 Continuum hypothesis and Martin’s axiom 05C55 Generalized Ramsey theory 05C65 Hypergraphs
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