Looking up definitions of the mathematical concept of three-dimensional space, the notion that it takes three independent real numbers to specify a point in it seems to be the defining property of the "three-dimensional" in "three-dimensional space". I was very surprised to learn however that there is a bijection between $\mathbb R$ and $\mathbb R^n$ for any finite $n$, such as $3$. In other words, it is possible to specify any point in three-dimensional space with a single real number, as could be done for any other finite-dimensional space.
This seems like a huge disconnect between how we perceive mathematics and the physical world to be connected, and the definition of three-dimensional space itself. What gives? What is then the fundamental property that makes three-dimensional space different from, say, two-dimensional space?
The bijection is ugly and not useful for anything; in particular it is badly discontinuous, hence not differentiable, so it can't be used e.g. to do differential geometry.
There is a theorem called invariance of domain guaranteeing that there is no bijection between $\mathbb{R}^n$ and $\mathbb{R}^m$ for $n \neq m$ which is continuous with continuous inverse (a homeomorphism). More generally it guarantees that the dimension of a topological manifold is well-defined. Understanding what exactly this all means requires understanding what a topology is.