  We study the topologically twisted string theory on the general back-ground
$AdS_3\times {\cal N}$ which is compatible with the world-sheet N=2
superconformal symmetry and is extensively discussed in the recent works
(hep-th/9904024, hep-th/9904040). After summarizing the algebraic structure of
the world-sheet topological theory, we show that the space-time (boundary)
conformal theory should be also topological. We directly construct the
space-time topological conformal algebra (twisted N=2 superconformal algebra)
from the degrees of freedom in the world-sheet topological theory. Firstly, we
work on the world-sheet of the string propagating near boundary, in which we
can safely make use of the Wakimoto free field representation. Secondly, we
present a more rigid formulation of space-time topological conformal algebra
which is still valid far from the boundary along the line of (hep-th/9903219).
We also discuss about the relation between this space-time topological theory
and the twisted version of the space-time N=2 superconformal field theory given
in (hep-th/9904024, hep-th/9904040).
