Brower and Kakutani fixed point theorems in definably complete structures

Let $\mathbb{K}$ be a definably complete expansion of a field, and let $X$ be a definable non-empty closed subset of some Cartesian power of $\mathbb{K}$. A fixed point of a map $f:X\to X$ is a point $x\in X$ such that $f(x)=x$.

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

The candidate is presented only as the names of two fixed point theorems, without a mathematical assertion or hypotheses in the supplied text. Its status and intended formulation therefore require checking against the source.