Quick question regarding the generally accepted definition for a space.
Suppose we consider the space $C_{0}^{\infty}(\Omega)$ of smooth functions of with compact support in $\Omega$, do we generally require strict containment of the support in $\Omega$, or do we allow the support to coincide with $\Omega$ in the case of a compact set $\Omega$ and the space to be equivalent to $C^{\infty} (\Omega)$.
The support of a function is closed, by definition.
Every closed subset of a compact set is compact.
So, if $\Omega$ is a compact set, the subscript is pointless. Just write $C^\infty (\Omega)$ then. Better yet, write $C^\infty(K)$ because the letter $K$ is typically associated with compact sets, while $\Omega$ is associated with open sets.
Generally, to work with notions of smoothness in a closed set, one assumes the existence of a larger open set on which smoothness holds.