On non-measurability of ℓ_{∞}/𝑐₀ in its second dual

We show that ℓ ∞ / c 0 = C ( N ∗ ) \ell _\infty /c_0=C(\mathbb {N}^*) with the weak topology is not an intersection of ℵ 1 \aleph _1 Borel sets in its Čech-Stone extension (and hence in any compactification). Assuming ( CH ), this implies that ( C ( N ∗ ) , w e a k ) (C(\mathbb {N}^*),\mathrm {weak}) has no continuous injection onto a Borel set in a compact space, or onto a Lindelöf space. Under ( CH ), this answers a question of Arhangel’skiĭ.