  We study basic properties of flow equivalence on one-dimensional compact
metric spaces with a particular emphasis on isotopy in the group of (self-)
flow equivalences on such a space. In particular, we show that an
orbit-preserving such map is not always an isotopy, but that this always is the
case for suspension flows of irreducible shifts of finite type. We also provide
a version of the fundamental discretization result of Parry and Sullivan which
does not require that the flow maps are either injective or surjective. Our
work is motivated by applications in the classification theory of sofic shift
spaces, but has been formulated to supply a solid and accessible foundation for
other purposes.
