Is there a way to represent the interior of a circle with a curve?
Solution 1:
Since topologically a disc and a square are the same, most of what you might want to know about this falls under the heading of Space-filling curves. To summarize, the answer to your main question is that the disc $D^2$ is the image of the interval $[0,1]$ under a continuous map, but not a one-to-one (non-intersecting) continuous map. So it depends on exactly what you mean by curve.