Despair of a technical detail

Question

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:

1. Why was this requirement not included in the official definition?
2. 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 $\omega(y) \neq \omega(x)$?
3. If "yes" for 2., can a worst-case of $\omega(y)\cap \omega(x) = \emptyset$ be achieved?

Answer 1

1. I don't know really.

2. 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 $x(t) \to \{x, y\}$.

3. 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.