I am reading Rudin Functional Analysis, there is one point he said (Remark 6.9, page 156) that it is obvious that $\mathcal{D}_K$ has empty interior relative to $\mathcal{D}(\Omega)$, can anyone explain it to me?
For each compact set $K\subset \Omega \subset \mathbb{R}^n$, we define $\mathcal{D}_K$ be the set of function $f$ smooth with support in $K$, i.e., $\mathcal{D}_K = C_c^\infty(K)$ while $\mathcal{D}(\Omega) = C_c^\infty(\Omega)$.
Pick $K$, a function $f \in \mathcal{D}_K$, and an open neighborhood $U$ of $f$ in the $\mathcal{D}(\Omega)$ topology. Now find a function $g \in U$ whose support is not contained in $K$. For example, if $h \in C_0^\infty(\Omega)$ is supported in $K' \subset \Omega$ and $K \cap K' = \emptyset$, then $g = f + \varepsilon h$ should work if $\varepsilon \ne 0$ is sufficiently small.
This shows that there are no open sets in the $\mathcal{D}(\Omega)$ topology that consist entirely of functions supported in a fixed compact subset $K$, which is what you want to show.