  Topological phases protected by symmetry can occur in gapped
and---surprisingly---in critical systems. We consider non-interacting fermions
in one dimension with spinless time-reversal symmetry. It is known that the
phases are classified by a topological invariant $\omega$ and a central charge
$c$. We investigate the correlations of string operators, giving insight into
the interplay between topology and criticality. In the gapped phases, these
non-local string order parameters allow us to extract $\omega$. Remarkably,
ratios of correlation lengths are universal. In the critical phases, the
scaling dimensions of these operators serve as an order parameter, encoding
$\omega$ and $c$. We derive exact asymptotics of these correlation functions
using Toeplitz determinant theory. We include physical discussion, e.g.,
relating lattice operators to the conformal field theory. Moreover, we discuss
the dual spin chains. Using the aforementioned universality, the topological
invariant of the spin chain can be obtained from correlations of local
observables.
