Despair of a technical detail Our setup (from the book Dynamical Systems and Chaos from Taken)is:

Then the authors propose an ad-hoc definition of the concept of an attractor:

But then they show that there are examples of $\omega (x)$ such that one can start arbitrarily closed to $\omega (x)$, but the orbit then goes first away and only then returns. This prompts them to give the second, official definition:

But now, oddly, the requirement that "a $V$ around $x\in M$ exists, such that for all $y\in V$ we have $\omega (y)=\omega (x)$", as in the ad-hoc definition, is missing.
Hence my questions:

*

*Why was this requirement not included in the official definition?

*Do there actually exist examples of attractors (in the sense of the official definition) such that no matter how small of a  $V$  around $x$ one might choose, there always exist $y\in V$ with $\omega(y) \neq \omega(x)$?

*If "yes" for 2., can a worst-case of $\omega(y)\cap \omega(x) = \emptyset$ be achieved?


*

*I don't know really.


*Consider the topology $\tau= \{ \{x,y\},  \{x\},\emptyset \}$ with the trivial stationary dynamics. Then $\omega(x)$ is an attractor. But then $y(t) \to y$ only, whereas $x(t) \to \{x, y\}$.


*Yes, in case $\omega(y) = \emptyset$ for example. See the photo below. Any point $x$ in the component 1 has the component (the circle minus the point) equal to $\omega (x)$. On the other hand, any point $y$ in the component 3 will have an empty $\omega (y)$ because it will (try to) tend to the missing point.
Notice that the covering (component 1) is constructed with the topologist's sine curve in mind.

Apologies in case I have messed something up.
