I'm currently doing some study into the area of space filling curves, specifically from Sagans text on the subject, and I'm struggling with the actual definition.
Sagan gives the definition that a curve is space filling if it is continuous and its image has strictly positive Jordan content, however in the text he then only goes on to show that the curves discussed are continuous and surjective to justify them being space filling.
I was hoping some help could be given to understand the intuition between linking these two ideas.
A curve is space filling if it is continuous and the image has positive Jordan content.
The square $[0,1]^2$ has positive Jordan content.
If a curve is continuous, its range is the square $[0,1]^2$, and it is surjective, then its image is the full square $[0,1]^2$. Thus its image has positive Jordan content and the curve is space filling.