Balleans of bounded geometry and G-spaces
Ігор Протасов · Matematychni Studii · 2008
A ballean (or a coarse structure) is a set endowed with some family of subsets which are called the balls.The properties of the family of balls are postulated in such a way that a ballean can be considered as an asymptotical counterpart of a uniform topological space.We prove that every ballean of bounded geometry is coarsely equivalent to a ballean on some set X determined by some group of permutations of X.Áîëåàí (èëè ýêâèâàëåíòíàÿ ñòðóêòóðà) ìíîæåñòâî îñíàùåííîå íåêîòîðûì ñåìåéñòâîì ïîäìíîæåñòâ, íàçûâàåìûõ øàðàìè.Ñâîéñòâà ñåìåéñòâà øàðîâ ïîñòóëèðóþòñÿ òàêèì îáðàçîì, ÷òî áîëåàí ìîæíî ðàññìàòðèâàòü êàê àñèìïòîòè÷åñêèé àíàëîã ðàâíîìåðíî òîïîëîãè÷åñêîãî ïðîñòðàíñòâà.Äîêàçûâàåòñÿ, ÷òî êàæäûé áîëåàí îãðàíè÷åííîé ãåîìåòðèè ãðóáî ýêâèâàëåíòåí áîëåàíó íåêîòîðîãî ìíîæåñòâà X, îïðåäåëÿåìîãî ñ ïîìîùüþ íåêîòîðîé ãðóïïû ïåðåñòàíîâîê X.A ball structure is called• lower symmetric if, for any α, β ∈ P , there exist α , β such that, for every• upper symmetric if, for any α, β ∈ P , there exist α , β such that, for every x ∈ X, B(x, α) ⊆ B * (x, α ), B * (x, β) ⊆ B(x, β );• lower multiplicative if, for any α, β ∈ P , there exists γ ∈ P such that, for every x ∈ X, B(B(x, γ), γ) ⊆ B(x, α) ∩ B(x, β);• upper multiplicative if, for any α, β ∈ P , there exists γ ∈ P such that, for every x ∈ X, B(B(x, α), β) ⊆ B(x, γ).