  We describe the modules in the Ziegler closure of ray and coray tubes in
module categories over finite-dimensional algebras. We improve slightly on
Krause's result for stable tubes by showing that the inverse limit along a
coray in a ray or coray tube is indecomposable, so in particular, the inverse
limit along a coray in a stable tube is indecomposable. In order to do all
this, we first describe the finitely presented modules over and the Ziegler
spectra of iterated one-point extensions of valuation domains. Finally we give
a sufficient condition for the $k$-dual of a $\Sigma$-pure-injective module
over a $k$-algebra to be indecomposable.
