We really cannot help you unless you give us a lot more information about the axioms and definitions used in your course. If I had to guess, I would say that this is from a set of notes based on class notes given a long time ago by R L Moore or one of his students.

For example: What does the first point to the right of M mean?

If my guess is correct, then it would imply that eachregionthat contains p must contain a point of M distinct from p. Otherwise p would not bethe first point to the right of M. That is enough to make p a limit point of M.

But you see that is just a wild guess as to what the terms mean.

So you are going to have to fill a great many more details to get any meaningful help.