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Abstract
We say that f: ℝ → ℝ is LIF if it is linearly independent over ℚ as a subset of ℝ2 and that it is a Hamel function (HF) if it is a Hamel basis of ℝ2. We construct an example of HF bijection and use a similar method to prove that any function can be represented as the composition of three HF’s as well as the limit of uniformly convergent sequence of HF’s. Finally we consider products of HF’s, maximal invariant classes (with respect to several algebraic operations) and pose some open problems concerning sets of continuity points of HF’s.
Abstract
We construct a translation invariant σ-ideal T(κ) (where κ is an infinite cardinal number) such that covt (T(κ)) = 2κ while cov (T(κ)) = cof (T(κ)) = ω1. The constructions can be carried out in R as well.
Abstract
We prove that axiom CPAgame 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.
representation of 𝔄 on a Banach space X with Hamel basis B . We claim that B is finite. Suppose on the contrary that B is not finite. Hence B contains an infinite set of linearly independent vectors { υ i | i ∈ ℕ}. Consider the descending chain I
