4,586 research outputs found
Ultrafilters on -spaces
For a discrete group and a discrete -space , we identify the
Stone-\v{C}ech compactifications and with the sets of all
ultrafilters on and , and apply the natural action of on
to characterize large, thick, thin, sparse and scattered subsets of
. We use -invariant partitions and colorings to define -selective and
-Ramsey ultrafilters on . We show that, in contrast to the
set-theoretical case, these two classes of ultrafilters are distinct. We
consider also universally thin ultrafilters on , the -points, and
study interrelations between these ultrafilters and some classical ultrafilters
on
On rapid idempotent ultrafilters
This short note contains the proofs of two small but somewhat surprising
results about ultrafilters on : 1. strongly summable ultrafilters
are rapid, 2. every rapid ultrafilter induces a closed left ideal of rapid
ultrafilters. As a consequence, there will be rapid minimal idempotents in all
models of set theory with rapid ultrafilters
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