Rudin argues in "functional analysis", the test function space $D(\Omega)$ on nonempty open set $\Omega$ is not metrizable. Let $D_K$ be the subspace of $D(\Omega)$ consisting of functions with support contained in compact $K$. Rudin said that "it is obvious that each $D_K$ has empty interior relative to $D(\Omega)$". I don't understand after thinking for hours. Any help is appreciated.