II. Definitions
1. The lower bound of an interval is the leftmost member of the interval. In the interval [r1, r2], r1 is the lower bound of the interval.
2. The upper bound of an interval is the rightmost member of the interval. In the interval [r1, r2], r2 is the upper bound of the interval.
3. Given the set, S = {1, 2, 3}. We define a closed interval of the set as IS = [1, 3] where both upper and lower bounds are included in the interval.
4. Given the set, S = {1, 2, 3}. We define an open interval of the set as IS = (1, 3) where the upper and lower bounds are excluded from the interval.
5. Given the set, S = {1, 2, 3}. We define a lower open interval of the set as IS = (1, 3] where the lower bound is excluded from the interval and the upper bound is included in the interval.
6. Given the set, S = {1, 2, 3}. We define an upper open interval of the set as IS = [1, 2) where the lower bound is included in the interval and the upper bound is excluded from the interval.
7. A conjoined interval pair is a pair of intervals where the upper bound of one and the lower bound of the other are the same member. [ri, [rk,] rj] is an example of a conjoined interval pair where rk is both the upper bound of [ri, rk], the lower bound of [rk, rj] and ri < rk < rj.
8. A relative bound is the number that is common to both intervals in a conjoined interval pair. In the conjoined interval pair [r1, [r3,] r2], r3 is the relative bound in both intervals [r1, r3] and [r3, r2].
9. An interval of a set may be partitioned by creating a conjoined interval pair per definition 7 and then splitting the conjoined interval pair into sub-intervals with the relative bound being the upper bound of one sub-interval and the lower bound of the other sub-interval.
Example:
S = {1, 2, 3}
IS = [1, 3] (IS is read the interval I on set S)
Partition IS as follows -
IS = [1, 3]
= [1, [2], 3]
= [1, 2], [2, 3]
10. When no sub-intervals can be further subdivided then the interval is called fully partitioned.
11. The immediate predecessor of a number λ is a number β such that there exists no number δ where
β < δ < λ.
12. The immediate successor of a number λ is a number β such that there exists no number δ where
λ < δ < β.
13. For any 2 real numbers λ and β in [r1, r2], we can always find another real number, δ, such that if λ > β then β < δ < λ and if λ < β then λ < δ < β. Therefore from definitions 11 and 12 we know that there are no immediate predecessors or successors of any of the elements of [r1, r2]; that is, [r1, r2] is a continuum.