Want to show every 0 -manifold is a countable and discrete space.
Let M be a 0-manifold. By definition, it is second countable, Hausdorff and each point in M has a neighborhood homeomorphic to $\mathbb{R}^{0}$. As a homeomorphism is a bijection, every point is an open set in M. Let A $\subset$ $M$. As $A = \bigcup_{x \in A}$ $\{$ $x$ $\}$. It follows that A is open, and similarly the complement of A is open , hence A is closed. As every set is both open and closed, M is a discrete space.
How do I show that M is countable?
Hint:
$M$ is second countable, so there is a countable base $\mathcal{U}$. However, you have shown that every singleton $\{x\}$ is an open set. Use the definition of base to show that $\{x\} \in \mathcal{U}$ for every $x \in M$, and conclude from this that $M$ is (at most) countable.
More generally, this shows that a second countable space can only have at most countably many isolated points.