We show that every definable nested family of closed and bounded subsets of a P-minimal field K has nonempty intersection. As an application we answer a question of Darnière and Halupczok showing that P-minimal fields satisfy the “extreme value property”: for every closed and bounded subset $U\subseteq K$ and every interpretable continuous function $f:U\to {\mathrm{\Gamma }}_K$ (where ${\mathrm{\Gamma }}_K$ denotes the value group), f(U) admits a maximal value. Two further corollaries are obtained as a consequence of their work. The first one shows that every interpretable subset of $K\times {\mathrm{\Gamma }}_K^n$ is already interpretable in the language of rings, answering a question of Cluckers and Halupczok. This implies in particular that every P-minimal field is polynomially bounded. The second one characterizes those P-minimal fields satisfying a classical cell preparation theorem as those having definable Skolem functions, generalizing a result of Mourgues.

我们证明了P最小域K的每个可定义的封闭和有界子集的嵌套族都具有非空交集。作为一个应用程序，我们回答了 Darnière 和 Halupczok 的问题，表明P最小域满足“极值属性”：对于每个封闭和有界子集$ü\subseteq ķ$和每一个可解释的连续函数$F：ü\to {\mathrm{\Gamma }}_{ķ}$（在哪里${\mathrm{\Gamma }}_{ķ}$表示值组），f ( U ) 允许最大值。由于他们的工作，我们得到了另外两个推论。第一个表明每个可解释的子集$ķ\times {\mathrm{\Gamma }}_{ķ}^n$已经可以用环的语言来解释，回答了 Cluckers 和 Halupczok 的问题。这尤其意味着每个P最小域都是多项式有界的。第二个将满足经典细胞制备定理的那些P最小场表征为具有可定义 Skolem 函数的那些，概括了 Mourgues 的结果。