Intersections of ℓp spaces in the Borel hierarchy

Abstract

We show that if Y is one of the spaces ℓq, c0, ℓ∞ or ⋂p>bℓp where 0<q,b<∞, and if the F-space ⋂p>aℓp is contained in Y properly (0⩽a<∞), then ⋂p>aℓp first shows up in the Borel hierarchy of Y at the multiplicative class of the third level. In particular ⋂p>aℓp is neither an Fσ nor a Gδ subset of Y. This answers a question by Nestoridis. This result provides a natural example of a set in the third level of the Borel hierarchy and with its help we also give some examples in the fourth level.