The Unit Interval as Arithmetic Interior

We define the set U = {1/n : n ∈ ℤ, n ≥ 2} under multiplication and characterise its algebraic structure. U is a commutative semigroup isomorphic to (ℤ≥2, ×) — the positive integers greater than or equal to 2 under multiplication. The irreducible elements of U are exactly the unit fractions at prime denominators, {1/p : p prime}, with every element admitting a unique factorisation into irreducibles corresponding directly to the prime factorisation of its denominator. The multiplicative identity (1) is external to U: it is not in U, and no finite product of elements of U equals 1. The supremum of U is 1/2. We note that the total ℓ² weight of U satisfies Σ_{x∈U} x² = ζ(2) − 1 = π²/6 − 1, connecting U to the Basel problem. These properties are direct consequences of the Fundamental Theorem of Arithmetic and standard semigroup theory, presented here as a short conceptual note whose purpose is to make the prime structure of ℤ≥2 geometrically visible in the unit interval. No new algebra claimed