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Let $U$ be a domain of type $H$ in a Brelot potential theory. A compact $K$ in $U$ is a $G_{\delta }$ in $U$ iff $U-K$ has at most countably many components. If $F$ is a relatively closed locally polar subset of $U$, any $G_{\delta }$ in $F$ is a $G_{\delta }$ in $U$. If $V$ is a domain in $U$, all Borel subsets of $\partial V\cap U$ are Baire even if $\partial V\cap U$ is not metrizable. The known results concerning equivalences between weak thinness, thinness, and strong thinness of a set $A$ at a point $x\notin A$ are extended from the case where $\left\{x\right\}$ is a $G_{\delta }$ to the cases in which $A$ meets only countably...