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

Burke, Dennis; Pol, Roman

doi:10.1090/S0002-9939-03-06983-1

On non-measurability of ${\ell _\infty }/c_0$ in its second dual
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by Dennis K. Burke and Roman Pol

Proc. Amer. Math. Soc. 131 (2003), 3955-3959

DOI: https://doi.org/10.1090/S0002-9939-03-06983-1

Published electronically: June 30, 2003

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Abstract:

We show that $\ell _\infty /c_0=C(\mathbb {N}^*)$ with the weak topology is not an intersection of $\aleph _1$ Borel sets in its Čech-Stone extension (and hence in any compactification). Assuming (CH), this implies that $(C(\mathbb {N}^*),\mathrm {weak})$ 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.

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