This is the final version of the article. Available from American Physical Society via the DOI in this record.It is shown theoretically that a one-dimensional crystal with time reversal symmetry is characterized by a Z_{2} topological invariant that predicts the existence or otherwise of edge states. This is confirmed experimentally through the construction and simulation of a photonic crystal analogue in the microwave regime.We thank an anonymous referee for helpful comments. The authors acknowledge financial support from the EPSRC through the QUEST program grant (Grant No. EP/I034548/1), C.A.M.B. was supported by QinetiQ and the EPSRC through the Industrial CASE scheme (Grant No. 08000346). H.M. is supported by a grant from the U. S. Department of Energy to the Particle Astrophysics Theory group at CWRU. T.J.A. is funded by the Research Corporation for Science Advancement through a Cottrell Award
