During evaluation of different special cases of Gaussian curvature for the surface of revolution in Ch. 7 (page 152) of Elementary Differential Geometry by Pressley, it comes to the case when $\sigma(u,v)=(\text{cos}\ u\ \text{cos}\ v,\text{cos}\ u\ \text{sin}\ v,u)$. The book claims that it is not a surface at all. Why it is not a surface? What is it then?
EDIT - I add the text from the book:
Was it "not a surface" or "not a smooth surface" or "not a closed surface"? This is the surface of rotation of the graph of $y = \cos z$ about the $z$-axis. It is singular at z= $\frac \pi 2 + k\pi$. But other than that it qualifies as a surface.