  The Kosterlitz-Thouless transition is studied from the representation of the
systems's ground state wave functions in terms of Matrix Product States for a
quantum system on an infinite-size lattice in one spatial dimension. It is
found that, in the critical regime for a one-dimensional quantum lattice system
with continuous symmetry, the newly-developed infinite Matrix Product State
algorithm automatically leads to infinite degenerate ground states, due to the
finiteness of the truncation dimension. This results in \textit{pseudo}
continuous symmetry spontaneous breakdown, which allows to introduce a
pseudo-order parameter that must be scaled down to zero, in order to be
consistent with the Mermin-Wegner theorem. We also show that the ground state
fidelity per lattice site exhibits a \textit{catastrophe point}, thus resolving
a controversy regarding whether or not the ground state fidelity is able to
detect the Kosterlitz-Thouless transition.
