\ for \ all \ M<\infty$$ For two infinite series of real numbers $\{a_n\}$, $\{b_n\}$ we write $a_n=o_{\omega}(b_n)$ if ${\lim^{\omega}}_i \frac{a_i}{b_i}=0$ and $a_n=\Theta_{\omega}(b_n)$ (respectively $a_n=\mathcal{O}_{\omega}(b_n)$) if $0 < {\lim^{\omega}}_i (\frac{a_i}{b_i})<\infty$ (respectively ${\lim_i^{\omega}} (\frac{a_i}{b_i})<\infty$).

Let $(X_n,d_n)$ be a metric space for every $n\in \mathbb{N}$. Let $\{e_n \}=:e$ be a sequence of points such that $e_n\in X_n$. Consider the set $\mathcal{F}$ of sequences $g=\{g_n \}$ where $g_n\in X_n$ such that $d_n(e_n,g_n)\leq C(g)$ $(C$ is a constant depending only on $g)$. Define an equivalence relation between elements of $\mathcal{F}$ as follows;

$f=\{f_n\}$ is equivalent to $h=\{h_n\}$ in $\mathcal{F}$ if and only if $\lim^{\omega}_id_i(f_i,h_i)=0$

The equivalence class of an element $\{g_n\}$ of $\mathcal{F}$ is denoted by $\{g_n\}^{\omega}$. The $\omega$-limit $\lim^{\omega}(X_n)_e$ is the quotient space of $\mathcal{F}$ by this equivalence relation and distance between two elements is defined as follows;

$Dist(f,g)=\lim^{\omega}d_i(f_i,g_i)$ for $f=\{f_i\}^{\omega}$, $g=\{g_i\}^{\omega}$ in $\lim^{\omega}(X_n,d_n)_e$.

An asymptotic cone $Con^{\omega}(X,e,l)$ for a metric space $(X,D)$, is an $\omega$-limit of metric space $X_n$ as above, where $X_n=X$ for all $n\in \mathbb{N}$, $d_n=\frac{D}{l_n}$ for a given non decreasing infinite sequence $l=\{l_n\}$ of positive real numbers and for a given sequence of points $e=\{e_n\}$ in $X$.

If $\{Y_n\}$ is a sequence of subsets of $X$ endowed with the 