Dimension of a unit square

Question

We all know the bijection between a unit square and a unit segment (see this link for example).

Since I can thus represent every point of the square with only one number, does it mean that the square is of dimension 1? We know it is of dimension 2, so where is the bug?

Answer 1

Points on the line and points in the plane have the same cardinality. In your link, the decription says that the unit square is defined via the open interval $I=(0, 1)$ such that the square, $S = I \times I$, with obvious bijection between $I$ and $I \times I$.

Answer 2

The 'bug' is that the specified bijection is not continuous. If you require a continuous mapping between the unit square and unit segment, you will find that that is not possible. THAT is where the dimensionality comes into play.