Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound.
Linear continua are particularly important in the field of topology where they can be used to verify whether an ordered set given the order topology is connected or not.
[1] Unlike the standard real line, a linear continuum may be bounded on either side: for example, any (real) closed interval is a linear continuum.
Examples in addition to the real numbers: Even though linear continua are important in the study of ordered sets, they do have applications in the mathematical field of topology.
Then if b1 and b2 are two upper bounds of D with b1 < b2, b2 will belong to D), D and its complement together form a separation on X.