On Jan 28, 2014, at 1:24 PM, Neighbor <firstname.lastname@example.org> wrote:
> About the "open ray": how can a finite straight-line have no boundary????
All sets of points have boundaries. The question is whether the boundary is part of the set of points or not.
Think about a number line. Now think about the set of all numbers greater than 4. (all the points to the right of 4 on the number line) Now tell me what is the smallest number in that set? (the point closest to 4 on the number line)
You can?t answer the question. Even though the set of all numbers greater than 4 has a lower boundary, which is 4, it isn?t in the set itself.
You defined such a set of points by saying that G is part of AG and that BG is all the points left over. BG has a boundary (G) but the boundary itself isn?t part of BG (by your choice). Thus, BG is an open segment at G. At B it is a closed segment because B is part of BG.
You have actually used examples of both types of sets (sets that include their boundary and sets that do not), but you haven?t realized that this is what you are doing. When you say that BG doesn?t include G (because you used that with AG), you have basically defined a set (BG) that doesn?t include its boundary. Such a set is called an ?open? set.