I am confused on the definition of curve and space filling curve in Chapter 1 of the book by Sagan. I think my confusion comes from notation. Let $\mathcal{I}:=[0,1]$, $\mathcal{Q}:=[0,1]^2$ and $J_n$ be the Jordan measure.
(1) Consider the definition of a curve given by the author: if $f:\mathcal{I}\rightarrow \mathbb{E}^n$ is continuous, then $$ f_{\star}(\mathcal{I}):=\{f(x) \in \mathcal{R}(f) \text{ s.t. }x \in \mathcal{I}\cap \mathcal{D}(f) \} \overbrace{=}^{\text{added by me}}\{f(x) \in \mathcal{R}(f) \text{ s.t. }x \in \mathcal{I}\} \overbrace{=}^{\text{added by me}}\mathcal{R}(\mathcal{f}) $$ is a curve, where $\mathcal{D}(f)$ is the domain of $f$ and $\mathcal{R}(f)$ is the range of $f$.
Hence, a curve is a set (countable or uncountable) of $n\times 1$ vectors representing the range of a continuous function $f$.
(2) Consider now the definition of a space-filling curve: If $f:\mathcal{I}\rightarrow \mathbb{E}^n$ with $n\geq 2$ is continuous and $J_n(f_{\star}(\mathcal{I}))>0$, then $f_{\star}(\mathcal{I})$ is called a space-filling curve. We say that $f$ generates a space-filling curve. (In order to have $J_n(f_{\star}(\mathcal{I})>0$ we need the curve to be an uncountable set I think ?).
(3) Suppose that the range of a continuous function $f:\mathcal{I}\rightarrow \mathbb{E}^2$ is $\mathcal{Q}$. Using (2), $f_{\star}(\mathcal{I})$ is a space-filling curve.
Question from (3): As $f_{\star}(\mathcal{I})$ is a space-filling curve and $f_{\star}(\mathcal{I})=\mathcal{Q}$, why can't we say that $\mathcal{Q}$ is a space-filling curve? I understand that it would be wrong of course because a space-filling curve is generated by a function but I think I'm missing some fundamental intuition behind the whole picture. Could you help me to understand better?
Intuition tells us that the continuous image of a one dimensional set should be, somehow, one-dimensional. E.g. if you look at the image of $t\mapsto (t,t)$ in Euclidean space, which is just a line.
From a measure theoretic view, it's volume is $=0$ when using a measure adapted to the dimension of the target space (i.e. a measure which relates, e.g., to some kind of area).
The concept of space-filling curve shows in some important respects that our intuition is wrong, namely with respect to the measure theoretic view on the one hand side (this is what is addressed by you item (2)) and also by a geoemetrical dimension point of view (this is what is addressed by (3), the continuous image of something one dimensional is two dimensional)
The point is that we always talk about the continuous image of a set which is kind of thin and get something wich is (contraintuitive) fat in some specific sense. This is why we don't just look $\cal{Q}$ which, as a set, is quite uninteresting in this regard.