  The global structure of 1 + 1 dimensional compact Universe is studied in
two-dimensional model of dilaton gravity.
  First we give a classical solution corresponding to the spacetime in which a
closed time-like curve appears, and show the instability of this spacetime due
to the existence of matters.
  We also observe quantum version of such a spacetime having closed timelike
curves never reappear unless the parameters are fine-tuned.
