# Is there a $\mathbb{Q}$-basis of $\mathbb{R}$ that is Borel?

A question that came to my mind yesterday:

Applying zorn's lemma (and hence, the axiom of choice), one can show that there must be a Hamel basis for the $$\mathbb{Q}$$-vector space $$\mathbb{R}$$, i.e. a subset $$H\subset \mathbb{R}$$, such that any real number $$x$$ can be uniquely expressed as a finite $$\mathbb{Q}$$-linear combination of elements of $$H$$. There is no way to construct such a set $$H$$, but nevertheless we are still able to describe some properties of $$H$$, such as that $$H$$ must be uncountable or that $$H$$ must contain at most one rational (two rationals wouldn't be linearily independent anylonger) or that $$H$$ must be totally disconnected (suppose not, then $$H$$ contains a proper interval and therefore more than one rational). So its seems that everey such $$\mathbb{Q}$$-bases of $$\mathbb{R}$$ must have a very odd structure.

Now my question is: of all the possible $$\mathbb{Q}$$-bases of $$\mathbb{R}$$, is there at least one that is tame enough to be a borel set? I mean there are after all strange structured borel sets out there and there are at least uncountably many $$\mathbb{Q}$$-bases of $$\mathbb{R}$$, but on the other hand, a $$\mathbb{Q}$$-basis of $$\mathbb{R}$$ seems to behave too strange to be a borel set..

So what do you think?

• Commented Apr 10, 2020 at 8:37

It is not possible for a Hamel basis of $$\mathbb{R}$$ as a $$\mathbb{Q}$$ vector space to be Borel. See Joel Hamkins' answer here: https://mathoverflow.net/questions/46063/explicit-hamel-basis-of-real-numbers, which also describes some consistency results about how close to Borel such a basis can be.

As a complement to Chris Eagle's answer, we can add at least a bit of regularity can be asked of a Hamel basis of $$\mathbb{R}$$ over $$\mathbb{Q}$$: By using Ramsey methods, it can be shown that there is a Cantor set of reals that are linearly independent over the rationals (See Kechris' Classical Descriptive Set Theory, Exercise 19.2). So, there are Hamel bases that include a perfect set.