Ok, maybe, but I think the big obstacle is how you would define the metric on the space of space filling curves of a space. Can you think of a way?
This is most likely a stupid question however it occurred to me whilst studying spaces. I was told that there can be "space filling curves", the class did not go into detail on these and I have not done any work on them myself, however I was thinking that if there were infinitely many space filling curves then is there some way which we could consider these curves as a space themselves? With the obvious extension that maybe there could be a space filling curve of a space of space filling curves of a space.