Subtle hyperplanes

We show that the countably-dimensional vector space $C_{00}$ of all sequences with finite support contains a convex cone $K$ that does not include straight lines and is closed Archiemedean but not closed in the Mackey topology $\tau$ corresponding to the duality $\langle C_{00}| F\rangle$, where $F$ is a hyperplane in the algebraic dual space $C_{00}^\#$.