Countable Discrete Topological Spaces

Question

I’d like to show that a countable discrete space X is locally Euclidean of dimension 0. I know that the way to do this is by showing that each point in X has a neighborhood homemorphic to $\mathbb{R}^{0}$. The problem is that I’m not quite sure what $\mathbb{R}^{0}$ means. I was able to show that every countable discrete space has a countable basis and is trivially Hausdorff. Hence showing that it is locally Euclidean of dimension 0 means that it is a 0 dimensional manifold.

Answer 1

$\mathbb R^n$ is the standard model of a real vector space of dimension $n$. Thus $\mathbb R^0$ is a $0$-dimensional real vector space, thus it only contains a single element.