  We prove that unstable dp-finite fields admit definable V-topologies. As a
consequence, the henselianity conjecture for dp-finite fields implies the
Shelah conjecture for dp-finite fields. This gives a conceptually simpler proof
of the classification of dp-finite fields of positive characteristic.
  For $n \ge 1$, we define a local class of "$W_n$-topological fields",
generalizing V-topological fields. A $W_1$-topology is the same thing as a
V-topology, and a $W_n$-topology is some higher-rank analogue. If $K$ is an
unstable dp-finite field, then the canonical topology is a definable
$W_n$-topology for $n = \operatorname{dp-rk}(K)$. Every $W_n$-topology has
between 1 and $n$ coarsenings that are V-topologies. If the given
$W_n$-topology is definable in some structure, then so are the V-topological
coarsenings.
