thinking about bijections
..of the unit interval.
the unit interval is the segment of the number line between 0 and 1, although any segment comes out to pretty much the same thing, and a bijection is a transformation of the interval so that every point is goes to a unique point and each point has exactly one mapped to it. basically the point (npi!) of all this is that if it's a bijection, it has an inverse; given this transformation you can make another which sends every point back to where it came from. You can't do that to a transformation if two points go to the same point - because you don't know which one to send it back to, or if some point doesn't have any points mapped to it - because you've got nowhere to send it back to.
Ways of making a bijection:
- Stretching out part of the interval, and squishing up some of the rest to make room for it.
This will be continuous, meaning that points which start fairly close together end up fairly close together. Its inverse is also continuous.
- Taking two or more pieces, and swapping them round.
This will be continuous everywhere except at the endpoints of the pieces taken out and replaced.
- Taking one piece (or the whole interval) and turning it around.
Again continuous except at the endpoints of the piece you moved.
- Any combination of the above
If you apply one bijection, then another to an interval, the combined transformation is another bijection. So you can combine as many as you like of these three types (in practice one of each type is enough) to get more complicated transformations.
- Something more interesting
Some transformations can't be made from a combination of these three types. In particular, if you want the transformation to be discontinuous at more than a finite set of points. These are the ones I'm interested in...