I am trying to prove the following assertion: Let $B$ be a simply-connected set, and let $B' \subsetneq B$ be a proper open connected subset. Then, there exists a point $x \in \partial B'$ of the boundary such that there is a basis of neighborhoods of $x$ in $B$ whose intersections with $B'$ are connected.
This seems like it should be true. For example, in the one-dimensional case, a proper open subinterval of an interval always has an endpoint satisfying this property, whereas for a circle, one can take the complement of a point $x \in S^1$, and the boundary point $x$ does not have the property in question.
I am happy if one takes the special case of $B$ a manifold, or even a product of intervals.