  The $\eta-\xi$ space-time is suggested by Gui for the quantum field theory in
1988. This paper consists of two parts. The first part is devoted to the
discussion of the global properties of the $\eta-\xi$ space-time. The result
contains a proof which asserts that the $\eta-\xi$ space-time is homeomorphic
to $\mathbb{C}^{*}\times \mathbb{C}^{*}\times \mathbb{C}^{2}$ by means of two
explicit maps, which shows that the $\eta-\xi$ space-time allows a Lie group
structure. Thus some transformation groups, one of which is isomorphic to the
Lorentz group in two dimensions, can be found. The other part of the paper is
the discussion about the embedding of some subspaces in the $\eta-\xi$
space-time. In particular, it is pointed out that the Euclidean space-time and
the Minkowskian space-time are linked in a way in the $\eta-\xi$ space-time
such that the tilde field appears naturally. In addition some formulae in the
$\eta-\xi$ space-time reappear in a more natural way.
