  Topological order can be found in a wide range of physical systems, from
crystalline solids, photonic meta-materials and even atmospheric waves to
optomechanic, acoustic and atomic systems. Topological systems are a robust
foundation for creating quantized channels for transporting electrical current,
light, and atmospheric disturbances. These topological effects are quantified
in terms of integer-valued invariants, such as the Chern number, applicable to
the quantum Hall effect, or the $\mathbb{Z}_2$ invariant suitable for
topological insulators. Here we engineered Rashba spin-orbit coupling for a
cold atomic gas giving non-trivial topology, without the underlying crystalline
structure that conventionally yields integer Chern numbers. We validated our
procedure by spectroscopically measuring the full dispersion relation, that
contained only a single Dirac point. We measured the quantum geometry
underlying the dispersion relation and obtained the topological index using
matter-wave interferometry. In contrast to crystalline materials, where
topological indices take on integer values, our continuum system reveals an
unconventional half-integer Chern number, potentially implying new forms of
topological transport.
