On hyperballeans of bounded geometry

Balleans, hyperballeans and ideals

“…If B = (X, P, B) is a ballean, the subballean X ♭ of exp B having as support the family of all non-empty bounded subsets of B was already defined and studied in [10]. Note that, B is connected (resp., unbounded) if and only if B ♭ is connected (resp., unbounded).…”

“…The ballean C(X, I), as well as its subballean M(X, I), having as support the ideal {g ∈ C X | supp g ∈ I} of the ring C X , will play a prominent role in the paper (note that M(X, I) \ {∅} coincides with (X ♭ I−ary )). 1 If X = N and I = F N , then M(X, I) is the Cantor macrocube defined in [10]. Motivated by this, the ballean M(X, I), for an ideal I on a set X, will be called the I-macrocube (o, shortly, a macrocube) in the sequel.…”

“…The number of connected components of exp(X I ) Recall that dsc(B) denotes the number of connected components of a ballean B. Clearly,(10) dsc(exp(B)) = dsc(exp * (B)) + 1 ≥ 2 for every non-empty ballean B. We begin with the following crucial observation.…”