We initiate the study of definable [Formula: see text]-topologies and show that there is at most one such [Formula: see text]-topology on a [Formula: see text]-henselian NIP field. Equivalently, we show that if [Formula: see text] is a bi-valued NIP field with [Formula: see text] henselian (respectively, [Formula: see text]-henselian), then [Formula: see text] and [Formula: see text] are comparable (respectively, dependent).As a consequence, Shelah’s conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field.We conclude by showing that Shelah’s conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is [Formula: see text]-henselian.

中文翻译：

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可定义的 V 型拓扑、Henselianity 和 NIP

我们开始研究可定义的 [Formula: see text]-topologies，并表明在 [Formula: see text]-henselian NIP 字段上最多有一个这样的 [Formula: see text]-topology。等效地，我们证明如果 [Formula: see text] 是具有 [Formula: see text] henselian（分别为 [Formula: see text]-henselian）的双值 NIP 字段，则 [Formula: see text] 和 [Formula :see text] 是可比的（分别是依赖的）。因此，Shelah 对 NIP 场的猜想暗示了对 NIP 场的亨斯连猜想。此外，后一个猜想对于任何承认具有 dp 最小余数域的亨斯估计值的域都得到证明。我们通过证明 Shelah 猜想等价于以下陈述：任何不包含在有限域的代数闭包中的 NIP 域是 [公式：见正文]-henselian。