Definition of the omega-limit set The $\omega$-limit set of the set  $B$ is defined as $\omega(B) = \bigcap_{n \geq 0} \overline{\bigcup_{k \geq n} T^k(B) }$, where $T$ is some continuous mapping.
What is the intuition behind this definition? I found this image in the Wiggins's book "Introduction to Applied Nonlinear Dynamical Systems and Chaos" and it shows that the $\omega$-limit set of point $x$ is a point $x_0$ on the curve $\gamma$):

To me, it is clear  that  $y \in \omega(B) $ iff there exists sequence  $x_j \in B$ and sequence of integers $n_j \rightarrow \infty$  such that $T^{n_j} x_j  \rightarrow y  $  as $j \rightarrow \infty$. But I don't completely understand the definition from beginning, why do we need union and then intersection? Thanks a lot in advance.
 A: Similar to real analysis, in set theory, we have the concept of the $\lim \sup S_i$ of a sequence of sets $S_i$.
In general,
$$\lim_{n\to \infty} \sup A_n := \bigcap_{n\geq1}\bigcup_{i\geq n} A_i$$
A helpful phrase for the members of $\lim \sup A_n$ (at least for me) is that $a \in \lim \sup A_n$ are members of the sequence that are appear infinitely often.
We need the union part because it's not guaranteed that the $A_i$ are a monotonic sequence of sets. If so, we normally can dispense with the union.
Now, on to your variant:
$$\omega(B) = \bigcap_{n \geq 0} \overline{\bigcup_{k \geq n} T^k(B) }$$
Here we are defining the $\omega-$limit as the $\lim \sup$ of the closure of all the values visited after at least $k$ applications of the transformation $T$ on the starting set $B$.
Comparing to the general $\lim \sup$ of a set, we have $A_i = T^k(B)$ with the added piece that we are taking the closure of the union -- I'm guessing to cover the case where the points in $B$ converge asymptotically to the $\omega-$limit (which I suspect is usually the case in real applications).
In the case of the drawing $\gamma$ appears to be limit set of the dynamical system. It is the set to which all point visit infinitely often and/or converge to.
