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We show that every Sobolev function in $W^{1,p}_{\textrm{loc}}(U)$ on a$p$-quasiopen set $U \subset {\bf R}^n$ with a.e.-vanishing $p$-fine gradientis a.e.-constant if and only if $U$ is $p$-quasiconnected. To prove this we usethe theory of Newtonian Sobolev spaces on metric measure spaces, and obtain thecorresponding equivalence also for complete metric spaces equipped with adoubling measure supporting a $p$-Poincar\'e inequality. On unweighted ${\bfR}^n$, we also obtain the corresponding result for $p$-finely open sets interms of $p$-fine connectedness, using a deep result by Latvala.

A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected sets