Examples of the Cantor-Bendixson process on the reals

A few months ago I was introducing some research students to the mysteries and delights or ordinal numbers, particularly ordinal iteration. To do that I was using the Cantor-Bendixson rank of closed subsets of various simple spaces. I needed some examoples to show that in the reals R these ranks exhaust the countable ordinals. I realized that I didn’t have any such examples to hand. (It is some time since I had the pleasure of teaching any of this stuff.) After rummaging around the literature for a while I couldn’t find what I was looking for, so I decided to sit down and sort out some examples for myself. (I admit I didn’t rummage for very long, hours rather than days. So may be exactly what I was looking for is out there somewhere. Anyway, doing the work myself probably did me more good than taking a job lot off the shelf.) This note is the result of that exercise. There is nothing very novel here, but it might be useful to some of you out there. I you find these notes useful, do let me know. I might be persuaded to expand them if enough people send me money. Also, if there are other collections of simple examples that I should have cited, again let me know.