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Ultrafilters on $$\omega$$ and atoms in the lattice of uniformities. I. (English) Zbl 0657.54022
The authors study atoms in the lattice of (covering-) uniformities (on a given set). Let us mention some of their results. (Recall that an atom in the lattice of uniformities on a set X is a uniformity $${\mathcal U}$$ such that the uniformly discrete uniformity $${\mathcal D}$$ on X (the least element of this lattice) is the only uniformity which is strictly finer than $${\mathcal U}.)$$
For every uniformity $${\mathcal U}\neq {\mathcal D}$$ on a set X there exists an atom $${\mathcal A}$$ refining $${\mathcal U}$$. However the lattice of uniformities on any infinite set is not atomic (i.e. some element of the lattice is not a supremum of atoms). There are two types of atoms: proximally non- discrete (those inducing a non-trivial proximity) and proximally discrete ones. Both types are related to ultrafilters.
Proximally non-discrete atoms can be described in the following way: A uniformity $${\mathcal A}$$ is a proximally non-discrete atom iff it is a copy of $${\mathcal S}_{{\mathcal F}}$$ for some ultrafilter $${\mathcal F}$$. (Let $${\mathcal F}$$ be an ultrafilter on a set X. Then all covers $${\mathcal C}_ F=\{\{x,1),(x,2)\}$$; $$x\in F\}\cup \{\{(x,y)\}$$; $$x\in X\setminus F\}\cup \{\{x,2)\}$$; $$x\in X\setminus F\}$$, where $$F\in {\mathcal F}$$, form a basis of the uniformity $${\mathcal S}_{{\mathcal F}}$$ on $$X\times \{1,2\}.)$$
This correspondence is used to classify ultrafilters or a countable set according as the corresponding atoms are, or are not, proximally fine. (A uniformity is proximally fine if it is the finest among all uniformities inducing its proximity.) Two typical results are: If $${\mathcal F}$$ is selective, then $${\mathcal S}_{{\mathcal F}}$$ is proximally fine, but under MA there exists an ultrafilter $${\mathcal F}$$ on $$\omega$$ which is not selective although $${\mathcal S}_{{\mathcal F}}$$ is proximally fine. (An ultrafilter $${\mathcal F}$$ on a set X is selective if for every partition $${\mathcal P}$$ of X either $${\mathcal F}\cap {\mathcal P}\neq \emptyset$$ or some $$F\in {\mathcal F}$$ is a selector for $${\mathcal P}$$ (i.e. $$| F\cap P| \leq 1$$ for all $$P\in {\mathcal P}).)$$ Under CH there exists an ultrafilter $${\mathcal F}$$ on $$\omega$$ such that $${\mathcal S}_{{\mathcal F}}$$ is proximally fine with respect to zero-dimensional uniformities but is not proximally fine. (A uniformity is zero-dimensional if it has a basis consisting of partitions.)
The relation of proximally discrete atoms to ultrafilters is provided by the next proposition. For every proximally discrete atom $${\mathcal A}$$ (in the lattice of uniformities on a set X) there exists a (unique) ultrafilter $${\mathcal F}$$ on X with $${\mathcal A}$$ refining $${\mathcal U}_{{\mathcal F}}.(The$$ so-called ultrafilter-uniformity $${\mathcal U}_{{\mathcal F}}$$ has all the covers $$\{$$ $$F\}$$ $$\cup \{\{x\}$$; $$x\in X\setminus F\}$$ (where $$F\in {\mathcal F})$$ as a basis.)
Ultrafilter-uniformities are seldom atoms, as the following result shows. Let X be a countable set (or, more generally), a set of non-measurable cardinality) and let $${\mathcal F}$$ be an ultrafilter on X. Then $${\mathcal U}_{{\mathcal F}}$$ is an atom (in the lattice of uniformities on X) iff $${\mathcal F}$$ is selective.