5 problems

Let $\mathbb{K}$ be a definably complete expansion of an ordered field, and let $f:[0,1]^2\to[0,1]^2$ be a definable continuous function. Then Brower's fixed-point conjecture. Ther…

Let $\mathbb{K}$ be an expansion of an ordered field, let $D \subseteq K_{\geq 0}$ be a definable closed discrete subset, let $h:D\to K_{\geq 0}$ be definable, and let…

Let $\mathbb{K}$ be a definably complete expansion of an ordered field, and let $\mu$ be the outer measure defined by infima of sums of lengths of definable open-interval covers in…

Brower and Kakutani fixed point theorem. Brower fixed point theorem. Kakutani fixed point theorem.

Pigeonhole principle. If $f$ is injective, then it is surjective.