Linear Continuum - Examples

Examples

• 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 reformulaton 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 = ) 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. 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.

If b doesn't belong to π₁(A), then b × 0 is the least upper bound of A, for if d < b, and d × e is an upper bound of A, then d would be a smaller upper bound of π₁(A) than b, contradicting the unique property of b.