Linear continuum

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In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.

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. More symbolically:

1. S has the least upper bound property, and
2. For each x in S and each y in S with x < y, there exists z in S such that x < z < y

A set has the least upper bound property, if every nonempty subset of the set that is bounded above has a least upper bound in the set. 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.