OK, this book is driving me nuts. This is exercise 16 from chapter 6.
For each set $E$, the set $P = E* ∪ (ℕ → E)$ of finite and infinite sequences from E is an inductive poset, under $⊆$.
For me, it's not clear what "$E* ∪ (ℕ → E)$" really means. E* is all possible words of all possible sizes from E, right? Isn't "$ℕ → E$" the same set as "$ℕ ⤫ E$"? Why would you want to join that with $E*$?
The solution does not make things more clear.
We claim that if $X ⊆ P$ is a chain [...], then the union set $⋃X$ is a function with domain some subset of $ℕ$. [...]
How could a union set have a domain?
Any enlightening remarks would be appreciated.
By the way, inductive poset means in this context a chain-complete partially ordered set.
[ The source appears to be p.256 of Notes on Set Theory, Yiannis Moschovakis —. MJD ]
$E^\star$ is the set of finite sequences of elements of $E$. For whatever reason, the book wants to consider the set of all sequences, both finite and infinite. So $E^\star$ does not do the job; it contains the finite sequences of elements of $E$, but not the infinite sequences.
An infinite sequence of elements of $E$ can be modeled as a function which, given a positive integer $n$, tells you the $n$th element of the infinite sequence. For example, supposing that $E$ is the set of all colors, then consider the sequence $\langle \text{red}, \text{blue}, \text{red}, \text{green}, \ldots\rangle$. This sequence can be identified with a function $f$ which has $$\begin{align} f(1) & = \text{red}\\ f(2) & = \text{blue}\\ f(3) & = \text{red}\\ f(4) & = \text{green} \\ & \vdots\end{align}$$
The reason we like to model sequences in this way is that we already have a lot of useful notation and theorems about the behavior of functions, and this model shows that we can understand and talk about a sequence as if it were a special kind of function.
This is what the notation $\Bbb N\to E$ means: it is the set of all functions that take an element of $\Bbb N$ and yield an element of $E$, and since we are identifying each such functions with a corresponding sequence of elements of $E$, the notation $\Bbb N\to E$ also means the set of all infinite sequences of elements of $E$.
The set $\Bbb N\times E$ is quite different from $\Bbb N\to E$: it is the set of all pairs where the first component is a positive integer and the second component is an element of $E$. If $E$ is the set of colors, then a typical element of $\Bbb N\to E$ is tabulated above, but a typical element of $\Bbb N\times E$ looks like $\langle 17, \text{purple}\rangle$. These pair objects are not at all like functions, so the elements of $\Bbb N\times E$ are quite different from the elements of $\Bbb N\to E$, and the sets are not at all alike.
Your other question, “how could a union set have a domain” is a little harder to answer without seeing the context, but I will try to explain. We model functions as sets of pairs. We can model this function: $$\begin{align} f(1) & = \text{red}\\ f(2) & = \text{blue}\\ f(3) & = \text{red}\\ f(4) & = \text{green} \\ & \vdots\end{align}$$
as a set of pairs that contains the pairs $\def\pr#1#2{\langle{#1},{#2}\rangle}\pr1{\text{red}},\pr2{\text{blue}}, \pr3{\text{red}}, \pr4{\text{green}}, $ and so on. (Your book explained this in sections 4.14–16; you might want to review it.) This allows us to understand functions as if they were sets, which allows us to take advantage of all the machinery and language for sets and use it to talk about functions. When a function is represented in this way, as a set of pairs, its domain is the set of all the first components of the pairs: $$\operatorname{domain} f = \{ x \mid \text{there is some $y$ such that } \pr xy \in f \}.$$ (On review, I see that this exact formula appears in section 4.15 in your book.) In the example above, the domain of $f$ contains $1,2,3,4,\ldots$; the domain of a sequence is always $\Bbb N$. Similarly the range of $f$ is the set of all second components of pairs from $f$, which in this case is some subset of $E$. Notice that doing things this way allows us to talk naturally about the range of a sequence, which would otherwise require a special definition. But because a sequence is a kind of function, and we know what the range of a function is, we can talk about the range of a sequence without having to remember any additional definitions or other machinery.
Probably the poset $P$ in your book is defined as a poset of sets of pairs, and then $X$ is some family of sets of pairs, and so its union is a set of pairs itself. This is what justifies treating $\bigcup X$ as a function, and speaking of its domain.
The tricky thing here, which is rarely pointed out, is that mathematical jargon conflates the model of the thing (the set of pairs) with the thing itself (the function) and will speak of them interchangeably; to a mathematician, the set of pairs and the function are the same thing, only with different aspects emphasized in each case. The situation with sequences is similar: in the correct context, a sequence of elements of $E$ is a function from $\Bbb N$ to $E$, and the two ideas are considered to be different aspects of the same thing. Mathematicians discussing the real line $\Bbb R$ will cheerfully refer to a real number, like $\pi$, as a “point”, and talk of subtracting one point from another, because the points and the numbers are the same thing. Mathematics gets a lot of its power from this approach, so it's important to be aware of it.