mathematics

Linear continuum

Linear continuum is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Linear continuum rather than just read about it. In short: 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 conditionally complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper…

Key takeaways

Linear continuum belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Linear continuum to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Linear continuum from memory before moving on to harder problems.

Reference excerpt

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 conditionally complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound in the set. 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.

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 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. 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 does not 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.

Non-examples The ordered set Q of rational numbers is not a linear continuum. Even though property b) is satisfied, property a) is not. Consider the subset A = {x ∈ Q | x < √2} of the set of rational numbers. Even though this set is bounded above by any rational number greater than √2 (for instance 3), it has no least upper bound in the rational numbers. (Specifically, for any rational upper bound r > √2, r/2 + 1/r is a closer rational upper bound; details at Methods of computing square roots § Heron's method.) The ordered set of non-negative integers with its usual order is not a linear continuum. Property a) is satisfied (let A be a subset of the set of non-negative integers that is bounded above. Then A is finite so it has a maximum, and this maximum is the desired least upper bound of A). On the other hand, property b) is not. Indeed, 5 is a non-negative integer and so is 6, but there exists no non-negative integer that lies strictly between them. The ordered set A of nonzero real numbers A = (−∞, 0) ∪ (0, +∞) is not a linear continuum. Property b) is trivially satisfied. However, if B is the set of negative real numbers: B = (−∞, 0) then B is a subset of A which is bounded above (by any element of A greater than 0; for instance 1), but has no least upper bound in A. Notice that 0 is not a bound for B since 0 is not an element of A. Let Z− denote the set of negative integers and let A = (0, 5) ∪ (5, +∞). Let S = Z− ∪ A. Then S satisfies neither property a) nor property b). The proof is similar to the previous examples.

Topological properties Even though linear continua are important in the study of ordered sets, they do have applications in the mathematical field of topology. In fact, we will prove that an ordered set in the order topology is connected if and only if it is a linear continuum. We will prove one implication, and leave the other one as an exercise. (Munkres explains the second part of the proof in ) Theorem Let X be an ordered set in the order topology. If X is connected, then X is a linear continuum. Proof: Suppose that x and y are elements of X with x < y. If there exists no z in X such that x < z < y, consider the sets:

A = (−∞, y) B = (x, +∞) These sets are disjoint (If a is in A, a < y so that if a is in B, a > x and a < y which is impossible by hypothesis), nonempty (x is in A and y is in B) and open (in the order topology), and their union is X. This contradicts the connectedness of X. Now we prove the least upper bound property. If C is a subset of X that is bounded above and has no least upper bound, let D be the union of all open rays of the form (b, +∞) where b is an upper bound for C. Then D is open (since it is the union of open sets), and closed (if a is not in D, then a < b for all upper bounds b of C so that we may choose q > a such that q is in C (if no such q exists, a is the least upper bound of C), then an open interval containing a may be chosen that doesn't intersect D). Since D is nonempty (there is more than one upper bound of D for if there was exactly one upper bound s, s would be the least upper bound. 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. This contradicts the connectedness of X.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Linear continuum

Start with the simplest possible case. Write down what Linear continuum claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Linear continuum before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Linear continuum ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Linear continuum

In research: Linear continuum appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Linear continuum in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Linear continuum is common in secondary-school and first-year university syllabi. It links to neighbouring topics Order theory, Topology, so understanding it makes those chapters shorter.
In everyday life: Look for Linear continuum outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Linear continuum in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Linear continuum means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Linear continuum out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Linear continuum in simple terms?

Why does Linear continuum matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Linear continuum?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Linear continuum.