Nice Hamel bases under the Covering Property Axiom

In the paper we prove that axiom CPA prism, which follows from the Covering Property Axiom CPA and holds in the iterated perfect set model, implies that there exists a Hamel basis which is a union of less than continuum many pairwise disjoint perfect sets. We will also give two consequences of this last fact. 1 The result and its consequences In this paper we will use standard set theoretic terminology as in [2]. We will consider the real line R as a linear space over the rationals Q. Any linear base of this space will be referred to as a Hamel base. For A ⊂ R we will write LIN(A) to denote the linear subspace of R spanned by A. ∗AMS classification numbers: Primary 26A15, 03E65; Secondary 26A30, 03E35.