# Properties of Hamel Bases

## Abstract

We recall that a Hamel basis is any base of the linear space (ℝ^{N}; ℚ; +; ·). We have constructed Hamel bases already many times in this book. Theorem 4.2.1 (cf., in particular, Corollary 4.2.1) asserts that there exist Hamel bases. More exactly (Lemma 4.2.1), for every set *A* ⊂ *C* ⊂ ℝ^{N} such that *A* is linearly independent over ℚ, and *E(C)* = ℝ^{N}, there exists a Hamel basis *H* of ℝ^{N} such that *A* ⊂ *H* ⊂ *C.* In particular, every set belonging to any of the classes A = B, ℭ, D(*D*), A_{ C }, B_{ C } contains a Hamel basis (Theorems 9.3.6 and 10.7.3 and Exercise 10.7). On the other hand, we have the following **Theorem 11.1.1.** *No Hamel basis belongs to any of the classes* A = B, C, D(*D*), A_{ C }, B_{ C }.

## Keywords

Measure Zero Choice Function Continuum Hypothesis Baire Property Hamel Basis## Preview

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