The pseudosphere in 2 and in n dimensions.

Question

According to Wikipedia, the pseudosphere is the surface of revolving a tractrix about its asymptote.

I have the following questions:

1. Does the half of the pseudosphere have a different name? I.e., the surface of revolving only the one "branch" of the tractrix about its asymptote.
2. The above surface (half of the pseudosphere?), as far as I understand, is a 2-dimensional surface, whose "base" is an 1-sphere. Does this generalise in higher dimensions? That is, does the n-pseudosphere have as its "base" an (n-1)-sphere?