Definition of $C^k( \overline{\Omega})$

Question

What is the exact definition of $C^k( \overline{\Omega})$ with $\Omega$ open set in $\mathbb{R}^n$? The functions in that space have domanin $\Omega$ or have domain $\overline{\Omega}$? Is there a general definition in manifold?

Answer 1

The definition I know is

$$ C^k(\overline{\Omega}) := \{ v\rvert_{\overline{\Omega}} \mid v \in C^k(U) \quad \text{for some open} \quad U \supset \overline{\Omega} \}. $$

I don't know about the manifold part.