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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 [...] such [...].

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.