  We present analysis of a single channel interacting quantum wire problem in
the presence of spin-orbit interaction. The spin-orbit coupling breaks the
spin-rotational symmetry from SU(2) to U(1) and breaks inversion symmetry. The
low-energy theory is then a two band model with a difference of Fermi
velocities $\delta v$. Using bosonization and a two-loop renormalization group
procedure we show that electron-electron interactions can open a gap in the
spin sector of the theory when the interaction strength $U$ is smaller than
$\delta v$ in appropriate units. For repulsive interactions, the resulting
strong coupling phase is of the spin-density-wave type. We show that this phase
has peculiar emergent topological properties. The gapped spin sector behaves as
a topological insulator, with zero-energy edge modes with fractional spin. On
the other hand, the charge sector remains critical, meaning the entire system
is metallic. However, this bulk electron liquid as a whole exhibits properties
commonly associated with the one-dimensional edge states of two-dimensional
spin-Hall insulators, in particular, the conduction of $2e^2/h$ is robust
against nonmagnetic impurities.
