If M is a manifold of dimension $ n \neq0$ then M has no isolated points.

Question

I am in doubt whether the following statement is true or false:

"If M is a manifold of dimension $ n \neq0$ then M has no isolated points."

The idea that made me find the true statement was as follows:

If $ p \in M $ is an isolated point, consider $ x: U \rightarrow \mathbb{R} ^ n $ a chart where $ U $ is open in M and $p \in U $. Since $ x $ is a homeomorphism and $ \{p \} $ is an open, we have $ \{x (p)\} $ is an open in $ \mathbb {R} ^ n $, but this is only possible if $ n = 0$, a contradiction.

Is that correct?

Answer 1

This was given as a comment, but the question needs closure, so I will answer it.

Yes.