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the sequences that extend σ by one element. Thus if A = { 0, 1 }, the first level of the tree consists of the sequences ⟨ 0 ⟩ and ⟨ 1 ⟩; the second level consists of the four sequences ⟨ 0, 0 ⟩, ⟨ 0, 1 ⟩, ⟨ 1, 0 ⟩, ⟨ 1, 1 ⟩; and so on. For each of the finite sequences σ in the tree, the set of all elements of A^ω that begin with σ is a basic open set in the topology on A. The open sets

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