Linear continuum
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 which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound. More symbolically:
A set has the least upper bound property, if every nonempty subset of the set that is bounded above 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.
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:
This map is known as the projection map. The projection map is continuous (with respect to the product topology on I × I) and is surjective. Let A be a nonempty subset of I × I which is bounded above. Consider π1(A). Since A is bounded above, π1(A) must also be bounded above. Since, π1(A) is a subset of I, it must have a least upper bound (since I has the least upper bound property). Therefore, we may let b be the least upper bound of π1(A). If b belongs to π1(A), then b × I will intersect A at say b × c for some c ∈ I. Notice that since b × I has the same order type of I, the set (b × I) ∩ A will indeed have a least upper bound b × c', which is the desired least upper bound for A.
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