The existence of a countable dense subset of a nonempty compact lattice of $\mathbb{R}^{n}$

Question

I'm reading about Tarski's theorem in the lecture note:

In the proof of Lemma 2.13, the author said that

Let $D=\left\{d_{1}, \ldots, d_{n}, \ldots\right\}$ be a (finite or) countable dense subset of $A$.

Could you please explain the existence of $D$? Thank you so much!

Answer 1

The solution invokes some basic facts from general topology.

1. $\mathbb R^n$ is separable which means that it contains a countable dense subset. "Countable" also includes finite.

2. A metrizable space is separable iff it is second countable which means that it has a countable base. In particular $\mathbb R^n$ is second countable (this can also be seen directly).

3. Any subspace of a second countable space is second countable.

Now $A$ is a subset of $\mathbb R^n$, thus $A$ is second countable. Since $A$ is metrizable, we find the desired $D$.