
Space Filling Curves
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.

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?

Indeed, this was my concern. We have a nice method of construction to work with, from the curves; I was wondering whether this could be in some way incorporated into a metric.

To do that you need to find out how the curves are different! Since they all fill up the same space, that is something they all share so in the end you cannot use this as a metric. However maybe there is some neat difference?