A ball structure is a triple \(\mathcal{B}=(X,P,B)\), where \(X\), \(P\) are non-empty sets, and for all \(x\in X\) and \(\alpha \in P\), \(B(x,\alpha)\) is a subset of \(X\) which is called a ball of radius \(\alpha\) around \(x\). Let \(A\subseteq X\). Let \(B^*(x,\alpha)=\{y \in X: x\in B(y,\alpha)\}\), \(B(A,\alpha)=\bigcup_{a\in A} B(a,\alpha)\).
A ball structure \(\mathcal{B}=(X,P,B)\) is a ballean if

(i)

for any \(\alpha, \beta, \in P\), there exists \(\alpha ', \beta ' \in P\) such that for every \(x\in X\), \(B(x,\alpha)\subseteq B^*(x,\alpha ')\), \(B^*(x,\beta)\subseteq B(x,\beta ')\);

(ii)

for every \(\alpha, \beta, \in P\), there exists \(\gamma \in P\) such that, for every \(x\in X\): \(B(B(x,\alpha), \beta)\subseteq B(x,\gamma)\);

(iii)

for any \(x,y\in X\) there exists \(\alpha \in P\), such that \(y\in B(x,\alpha)\).

A ballean is bounded if its support set \(X\) is bounded. Let \(X^b\) be the family of all non empty bounded subsets of \(X\), consider a new ballean \(\mathcal{B}^b=(X^b,P,B^b)\), where \(B^b(Y,\alpha)=\{Z \in X^b: Z\subseteq B(Y,\alpha), Y\in B(Z,\alpha)\}\). Let \(\mathcal{F}_X\) be the set of all finite subsets of \(X\). Let \(\mathbf{ F}_X\) be the ballean \((X, \mathcal{F}_X, B_\mathbf{F})\) where \[ B_\mathbf{F}= \begin{cases} \{x\} & x\notin F\\ F & x\in F \end{cases} \] The authors prove the following theorems:
{Theorem 2.1.} For an unbounded ballean \(\mathcal{B}\) the following statements hold:

(i)

\(\mathcal{B}^b\) is uniformly bounded locally finite if and only if \(\mathcal{B}=\mathbf{F}_X\);

(ii)

\(\mathcal{B}^b\) is of bounded geometry if and only if there exists a large subset \(Y\) of \(X\) such that \(\mathcal{B}_Y=\mathbf{F}_X\).

\(\mathcal{B}^b\) is of bounded geometry if there exists \(\alpha \in P\) and a function \(f:P\rightarrow \mathbb{N}\), such that for a subset \(S\subseteq B(x,\beta)\) such that \(B(x,\alpha)\cap S=\{x\}\) for each \(x\in S\), then \(|S|\leqslant f(\beta)\). \(\mathcal{B}^b\) is uniformly bounded locally finite, if for every \(\beta\in P\), there is a \(n(\beta)\in \mathbb{N}\) such that \(\vert B(x,\beta)\vert \leqslant n(\beta)\) for every \(x\in X\). A ballean \(\mathcal{B}=(X,P,B)\) is called asymptotically scattered if for every unbounded subset \(Y\) of \(X\), there is \(\alpha \in P\), such that for every \(\beta \in P\), there exists \(y\in Y\) such that \((B(y,\beta)\setminus B(y,\alpha))\cap Y=\emptyset\). Let \(\kappa\) be a cardinal, and \(\mathbf{Q}_{\kappa}\) be the ballean with support \(\mathbf{Q}_{\kappa}=\{(x_{\alpha})_{\alpha< \kappa}: x_{\alpha}\in [0,1], x_{\alpha} =0 \text{ for all but finitely many } \alpha < \kappa \},\) the set of radii \(\mathcal{F}\kappa \), and the balls \(B_{\mathbf{Q}}((x_\alpha)_{\alpha < \kappa}, F)=\{(y_{\alpha})_{\alpha< \kappa}: x_\alpha=y_\alpha \text{ for all } \alpha \in \kappa \setminus F\}\). \(\mathbf{Q}_{\omega}\) is known as the Cantor macrocube.
{Theorem 2.2}
Let \(\kappa \) be an infinite cardinal, \(n\in \mathbb{N}\), \([\kappa]^n=\{F\subset \kappa : |F|=n\}\), \(x\in \kappa\). Then the following statements hold:

(i)

The subballean of \(\mathbf{F}^b_{\kappa}\) with the support \([\kappa]^n\) is asymptotically scattered;

(ii)

The subballean of \(\mathbf{F}^b_{\kappa}\) with the support \(\{F \in \mathcal{F}_{\kappa}: x\in F\}\) is asymorphic to \(\mathbf{ Q}_{\kappa}\);

(iii)

\(\mathbf{F}^b_{\omega}\) can be partitioned into countably many pairwise close Cantor macrocubes but \(\mathbf{F}^b_{\omega}\) is not coarsely equivalent to \(\mathbf{Q}_{\omega}\).