Let $X\subseteq\mathbb{R}$, and $\forall x\in X, \{y\in X:y>x\}$ is countable. Prove $X$ is countable.

Question

Let $X\subseteq\mathbb{R}$, and $\forall x\in X, \{y\in X:y>x\}$ is countable.

Prove $X$ is countable.

I was trying to use that if $X$ is uncountable, it embeds into $\omega_1^*$ which is a contradiction, but I don't know how to show this embedding.

Answer 1

Hint Split the problem in two cases:

Case 1 $X$ is bounded from below.

Let $a= \inf(X)$. If $a \in X$ the claim is obvious. If $a \notin X$ then for each $n$ tehre exists some $a<a_n <a+\frac{1}{n}$.

Show that $$X= \bigcup_n \{y\in X:y>a_n\}$$

Case 2: $X$ is unbounded from below. For each $n$ pick some $a_n \in X$ such that $a_n <-n$.

Show that $$X= \bigcup_n \{y\in X:y>a_n\}$$