I have been reading the book "Introduction to Set Theory" by Jech and Hrbacek and have come to an exercise I'm having difficulty with in the chapter on filters and ultrafilters. The exercise comes from section 3 of the chapter on closed, unbounded, and stationary sets.
The exercise goes :
If $X$ is an unbounded subset of $\omega_{1}$, then the set of all countable limit points of $X$ is a closed unbounded set.
Before the exercise is the text :
If $X$ is a set of ordinals, then $\alpha$ is a limit point of $X$ if for every
$\gamma < \alpha$ there is $\beta \in X$ such that $\gamma < \beta < \alpha$. A countable $\alpha$ is a limit point of $X$ if and only if there exists a sequence
$\alpha_{0} < \alpha_{1} < \dots$ in $X$ such that $\sup\left( \{ \alpha_{n} \; \mid \; n \in \omega \} \right) = \alpha$. Every closed unbounded $C \subseteq \omega_{1}$ contains all its countable limit points.
$\newcommand{\Suc}{\text{Suc}}
\newcommand{\Lim}{\text{Lim}}
\newcommand{\Ord}{\text{Ord}}$
Here I use the notation $\Suc$ for the successor ordinals, $\Lim$ for the limit
ordinals, and $\Ord$ for the class of all ordinals.
Here is my attempt at a solution thus far :
Let :
\begin{equation}
A = \{ \alpha \in \Ord \; \mid \; \alpha \text{ is a limit point of } X \}
\end{equation}
First show :
\begin{equation}
\alpha \in A \Rightarrow \alpha \not \in \Suc
\end{equation}
Let's suppose :
\begin{equation}
\alpha \in A \text{ and } \alpha \in \Suc
\end{equation}
Let :
\begin{equation}
\alpha = \gamma + 1
\end{equation}
So :
\begin{equation}
\gamma < \alpha \text{ and } \not \exists \beta \in \Ord \text{ s.t. } \gamma < \beta < \alpha
\end{equation}
and since $X \subseteq \Ord$ :
\begin{equation}
\gamma < \alpha \text{ and } \not \exists \beta \in X \text{ s.t. } \gamma < \beta < \alpha
\end{equation}
So $\alpha$ cannot be a limit point of $X$ when $\alpha \in \Suc$. $\checkmark$
So :
\begin{equation}
\alpha \in A \Rightarrow \alpha \not \in \Suc \; \checkmark
\end{equation}
and :
\begin{equation}
A = \{ \alpha \in \Lim \; \mid \; \alpha \text{ is a limit point of } X \}
\end{equation}
We know since $X$ is unbounded :
\begin{equation}
\sup(X) = \omega_{1}
\end{equation}
We know :
\begin{equation}
\alpha \in X \Rightarrow \exists \beta \in A \text{ s.t. } \alpha \leq \beta
\end{equation}
Since every $\beta \in A$ is the supremum of some increasing sequence of length
$\omega$ in $X$.
So :
\begin{equation}
\sup(X) \leq \sup(A)
\end{equation}
and since :
\begin{equation}
\alpha \in A \Rightarrow \alpha < \omega_{1}
\end{equation}
We know :
\begin{equation}
\sup(A) \leq \omega_{1}
\end{equation}
So :
\begin{equation}
\omega_{1} = \sup(X) \leq \sup(A) \leq \omega_{1} \Rightarrow \sup(A) = \omega_{1} \; \checkmark
\end{equation}
and $A$ is unbounded. $\checkmark$
I am not sure how to show that $A$ is also closed. Can someone help with this ?
Best Answer
Closed means that any limit of elements of $A$, are also in $A$, so let's take increasing a sequence $(α_i ∈ A)_{i∈ω}$, and we will show that $\lim α_i=α∈ A$ (I assume monotocity, so $\lim$ here is the same as $\sup$).
We know that each $α_i$ is a limit of some increasing sequence $(β_i^j ∈ X)_{j\in\omega}$, define $f:ω→ω$ to be $f(i)=\min(j\in\omega\mid β_{i+1}^j>α_i)$ (note, it is well defined because $α_i=\lim_j β_i^j<\lim_j β_{i+1}^j=α_{i+1}$, also note that $f$ is not necessarily increasing).
I claim that $\lim_i β_{i+1}^{f(i)}=α$, indeed $β_{i+1}^{f(i)}<α_{i+1}<α$ for all $i$, so $\lim β_{i+1}^{f(i)} ≤α$, but if $x<α$, there exists $i$ such that $x<α_i<α$, and so $x<α_i<β_{i+1}^{f(i)}<α$, hence $\lim_i β_{i+1}^{f(i)}=α$.
So, given any limit point $α$ of $A$, we have a (countable) sequence from $X$, whose limit is $α$, hence $α∈A$