Space filling curves: for now, we’ll just work on continuous functions .
A curve is typically defined as a continuous function where is, say, a manifold (a 2’nd countable metric space which has neighborhoods either locally homeomorphic to or . Note: though we often think of smooth or piecewise linear curves, we don’t have to do so. Also, we can allow for self-intersections.
However, if we don’t put restrictions such as these, weird things can happen. It can be shown (and the video suggests a construction, which is correct) that there exists a continuous, ONTO function ; such a gadget is called a space filling curve.
It follows from elementary topology that such an cannot be one to one, because if it were, because the domain is compact, would have to be a homeomorphism. But the respective spaces are not homeomorphic. For example: the closed interval is disconnected by the removal of any non-end point, whereas the closed square has no such separating point.
Therefore, if is a space filling curve, the inverse image of a points is actually an infinite number of points; the inverse (as a function) cannot be defined.
And THAT is where this article and video goes off of the rails, though, practically speaking, one can approximate the space filling curve as close as one pleases by an embedded curve (one that IS one to one) and therefore snake the curve through any desired number of points (pixels?).
So, enjoy the video which I got from here (and yes, the text of this post has the aforementioned error)