  Let $(R,\mathfrak{m})$ be a commutative local noetherian ring. For an
artinian $R$-module $M$, we show the equality
$$\mathrm{cosupp}_RM=\mathrm{Cosupp}_RM$$ using the semi-discrete linearly
compactness of $R$-module $\mathrm{Hom}_R(R_\mathfrak{p},M)$ where
$\mathfrak{p}$ is a prime ideal of $R$.
