Definition of dimension

Question

Let us consider Euclidean space $\mathbb{R}^n$. We say it is $n$-dimensional because each vector in it is an $n$-tuple $(x_1,...,x_n)$. However, it is possible to represent this exact same space using only a single number by creating a bijection $\mathbb{R}^n\to\mathbb{R}$ (e.g. for $n=3$). Therefore, by the conventional definition of dimension, we must conclude that $\mathbb{R}^n$ is in fact one-dimensional.

How do we resolve this? What is a rigorous definition of dimension?

This question was inspired by this one.

Answer 1

You can resolve this when you notice that any such bijection does not preserve the vector space structure of $\Bbb R^3$. In simpler words, it is not a linear transformation.

(Assuming the axiom of choice, you can find such bijection which preserves the additive structure, and therefore the structure of a vector space over $\Bbb Q$, but never as a vector space over $\Bbb R$.)

The point is that cardinality "shakes off" disregards the structure and only considers the underlying set. But dimension requires structure, so you can't just define the dimension of a set as a vector space or a topological space, without specifying the topology or the vector space structure as well.

What is true is that $\Bbb R$ can be endowed with a $3$-dimensional real vector space structure. But this structure is incompatible with the usual structure of $\Bbb R$ as an ordered field.

Answer 2

For vector spaces, the definition of dimension is the number of vectors in a basis for the vector space.

A basis for a vector space V over a field $\mathbb{F}$ is a set of vectors such that every vector in V is a unique linear combination of the basis vectors. That is, $\{ v_1, ..., v_k \}$ is a basis when every vector $v \in V$ can be written uniquely as $v = a_1 v_1 + \dots + a_k v_k$ for scalars $a_1$, \dots, $a_k \in \mathbb{F}$.

It is possible to prove all bases for the same vector space have the same number of vectors, and this number is defined to be the dimension.

For $\mathbb{R}^n$, a basis is $(1,0,\dots,0)$, $(0,1,\dots,0)$, ..., $(0,0,\dots,1)$ so it has dimension $n$.