Thin Ultrafilters

Petrenko, Oleksandr; Протасов, Ігор

doi:10.1215/00294527-1626536

Cited by 6 publications

(5 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the next theorem, we summarize the main results from [8]. A free ultrafilter U on ω is called a T ℵ 0 -point if, for each minimal well ordering < of ω, there is a d < -thin member of U, where d < is the natural metric on ω defined by <.…”

Section: Thin Ultrafiltersmentioning

confidence: 99%

Ultrafilters on $G$-spaces

Petrenko,

Protasov

2015

Preprint

View full text Add to dashboard Cite

For a discrete group G and a discrete G-space X, we identify the Stone-Čech compactifications βG and βX with the sets of all ultrafilters on G and X, and apply the natural action of βG on βX to characterize large, thick, thin, sparse and scattered subsets of X. We use Ginvariant partitions and colorings to define G-selective and G-Ramsey ultrafilters on X. We show that, in contrast to the set-theoretical case, these two classes of ultrafilters are distinct. We consider also universally thin ultrafilters on ω, the T -points, and study interrelations between these ultrafilters and some classical ultrafilters on ω.

show abstract

Section: Thin Ultrafiltersmentioning

confidence: 99%

Ultrafilters on $G$-spaces

Petrenko,

Protasov

2015

Preprint

View full text Add to dashboard Cite

show abstract

“…By [5] and [7], this is so if c ℵ 2 . By [8,Proposition 4], under CH, there exist a T -point which is neither Q-point nor weak P -point. It is an open question [9, p. 348] whether T -points exist in ZFC without additional assumptions.…”

Section: Around T -Pointsmentioning

confidence: 99%

Asymptotic structures of cardinals

2014

View full text Add to dashboard Cite

A ballean is a set X endowed with some family F of its subsets, called the balls, in such a way that (X, F) can be considered as an asymptotic counterpart of a uniform topological space. Given a cardinal κ, we define F using a natural order structure on κ. We characterize balleans up to coarse equivalence, give the criterions of metrizability and cellularity, calculate the basic cardinal invariant of these balleans. We conclude the paper with discussion of some special ultrafilters on cardinal balleans.2010 MSC: 54A25; 05A18.

show abstract

“…For thin subsets of a group, their applications and modifications, see [9][10][11][12][13][14][15][16][17][18][19]. By [6] (Proposition 5) and [20] (Theorems 3 and 10), for a subset A of a group G, the following statements are equivalent:…”

Section: Introductionmentioning

confidence: 99%

Scattered Subsets of Groups

2015

Self Cite

View full text Add to dashboard Cite

We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that µ(A) = 0 for each left invariant Banach measure µ on G. It is also shown that every infinite group can be split into @0 scattered subsets.

show abstract

Thin Ultrafilters

Cited by 6 publications

References 9 publications

Ultrafilters on $G$-spaces

Ultrafilters on $G$-spaces

Asymptotic structures of cardinals

Scattered Subsets of Groups

Contact Info

Product

Resources

About