One naive way to define a topology on test functions ${\mathcal D}(\Omega)$ would be to exhaust $\Omega$ by compacts $(K_n)$ and to take the metric induced by the semi-norm system $$ {\| f \|} _ {n} := \| f \|_{C^n(K _ {n} )}, $$ i.e. $$ d(f, g) = \sum _ n 2^{-n} \frac{ \|f-g\| _ {n} }{ 1+\|f-g\| _ {n} } $$ I read (without any reference) that this yields a non-complete space.

Do you know a reference or a concrete example how to show non-completeness?