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Concept
Linear continuum
Summary
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:
S has the least upper bound property, and
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.
Unlike the standard real line, a linear continuum may be bounded on either side: for example, any (real) closed interval is a linear continuum.
The ordered set of real numbers, R, with its usual order is a linear continuum, and is the archetypal example. Property b) is trivial, and property a) is simply a reformulation of the completeness axiom.
Examples in addition to the real numbers:
sets which are order-isomorphic to the set of real numbers, for example a real open interval, and the same with half-open gaps (note that these are not gaps in the above-mentioned sense)
the affinely extended real number system and order-isomorphic sets, for example the unit interval
the set of real numbers with only +∞ or only −∞ added, and order-isomorphic sets, for example a half-open interval
the long line
The set I × I (where × denotes the Cartesian product and I = [0, 1]) in the lexicographic order is a linear continuum. Property b) is trivial. To check property a), we define a map, π1 : I × I → I by
π1 (x, y) = x
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.
Official source
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