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Definition

Given a sequentially Cauchy complete Archimedean ordered field $\mathbb{R}$, a function $f:\mathbb{R} \to \mathbb{R}$ is strongly extensional if for every $x:\mathbb{R}$ and $y:\mathbb{R}$, $\vert f(x) - f(y) \vert \gt 0$ implies that $\vert x - y \vert \gt 0$.

In particular, this definition applies to Dedekind real numbers, which is used for proving Brouwer's theorem? in real-cohesive homotopy type theory.

See also

• axiom R-flat

• Brouwer's theorem?

References

• Mike Shulman, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (arXiv:1509.07584, doi:10.1017/S0960129517000147)