Structure of homogeneous chains

Especially, for any pair of elements a, b of X, the set of elements between (properly) a and b is an interval of X, which we ca31 an open interval (a, b). The set oϊ upper bounds and the set of lower bounds of an element a of X, excluding the element a, are also called (unbounded) open intervals, and are denoted by (a,-) and (-, a) respectively. When two elements a and b are adjoined to the open interval (a, bj, we call it a closed interval [a, bl [a, b) denotes the interval (a, b)with adjoined a only, (a, b], [a, ) , and (-, a] are similarly defined*,