By Peano's space-filling curve, there exists a continuous map f: I -> I^2 whos image fills up the entire square I^2 (where I=[0, 1]). This can also be represented by gluing points of I together. Which points of I get glued together? I was looking at the proof of Peano's space-filling curve and I kind of get the idea it's all of them, although I'm having trouble justifying that. Would anyone like to shed some light on this?