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Math and science::Analysis::Tao::05. The real numbers

Cut property for real numbers

Cut property for real numbers

If \( A \) and \( B \) are nonempty, disjoint sets with \( A \cup B = \mathbb{R} \) and \( a < b \) for all \( a \in A \) and \( a < b \), then there exists a \( c \in \mathbb{R} \) such that \( x \le c \) whenever \( x \in A \) and \( x \ge c \) whenever \( x \in B \).

This a Dekekind cut and Dedekind completeness.

The cut property expresses the idea that sets of reals always have a boundary which is itself a real.

Cut property is equivalence to the idea of supremum

The cool thing about the cut property is that it that it is equivalent to 'Cauchy completeness', which is what Tao covered (every Cauchy sequence of real numbers converges to a real number). In addition, both of these are logically equivalent to the existence of supremum & infimum for sets of reals. The cut property feels a bit more tangible compared to the supremum & infimum, and it can be good to remember that the property encapsulates the same power.

In "Understanding Analysis", Stephen Abbott introduces the existence of supremums and infimums as an axiom which imbues the reals with completeness. He points out that the cut property instead could have been given as an axiom, and the existence of the supremum and infimum would then be implied.


Abbott, p19