Category:Continuum Property

This category contains pages concerning Continuum Property:

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.