Continuum Property

Theorem

The continuum property (of the set of real numbers $\R$) is a complementary pair of theorems whose subject is the real number line:

Least Upper Bound Property

Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded above.

Then $S$ admits a supremum in $\R$.

This is known as the least upper bound property of the real numbers.

Greatest Lower Bound Property

Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded below.

Then $S$ admits an infimum in $\R$.

This is known as the greatest lower bound property of the real numbers.

Also presented as

The Continuum Property can also be stated as:

The set $\R$ of real numbers is Dedekind complete.

Also known as

The Continuum Property of $\R$ is also known as:

the completeness axiom

the completeness property

the completeness postulate

Also see

• Dedekind's Theorem
• Definition:Dedekind Complete
• Continuum Property implies Well-Ordering Principle

Not to be confused with:

• The Continuum Hypothesis

Sources

• 1947: James M. Hyslop: Infinite Series (3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 3$: Bounds of a Function: Theorem $\text{A}$
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.4$: The Continuum Property