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A topological insulator^,,,,,,,, is a new state of quantum matter that is characterized by a finite energy gap in the bulk and gapless modes flowing along the boundaries that are robust against disorder scattering. The topological protection of the surface state could be useful for both low-power electronics and error-tolerant quantum computing^,. For a thin slab of three-dimensional topological insulator, the boundary modes from the opposite surfaces may be coupled by quantum tunnelling, so that a small, thickness-dependent gap is opened up^,,. Here we report such results from angle-resolved photoemission spectroscopy on Bi₂Se₃ films of various thicknesses grown by molecular beam epitaxy. The energy gap opening is clearly seen when the thickness is below six quintuple layers. The gapped surface states also exhibit sizeable Rashba-type spin–orbit splitting because of the substrate-induced potential difference …