Let $T$ be a topological semigroup, not necessarily discrete, and $\varphi:T\times X\to X$, where we denote by $(T, X) $, be semiflow on topological space $X$. This means that $\varphi$ is continuous and $\varphi(t_0, \varphi(t_1, x))=\varphi(t_0t_1, x)$ , we denote with $t_0(t_1x)=(t_0t_1)x$ for all $t_0, t_1\in T$ and all $x\in X$.
The set of limit points of $Tx=\{tx:t\in T\}$ is called omega limit set of $x$ and denote by $\omega_T(x)$. This definition is not clear for me.
In the some paper, Authors defined $$\omega_T(x)=\bigcap_{s\in T}\overline{Tsx}$$ But also, another Authors said that $y\in \omega_T(x)$ iff for every open set $U$ of $x$ in $X$ and every compact set $K$ in $T$, there is $t\in (T-K)$ with $tx\in U$.
Can you help me to know definition of $y\in\omega_T(x)$?