# Bases of open sets with connected intersections

I'm interested in knowing classes of topological spaces $X$ which admit a basis of open sets $\{U_i\}_{i\in I}$ such that $U_i\cap U_j$ is connected for all $i,j\in I$. Do manifolds have this property? Riemannian ones maybe? If not, what if we relax the condition by saying that $U_i\cap U_j$ has a finite number of connected components for $i,j\in I$?

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I reckon you consider the empty subspace to be connected (since for a Hausdorff space, at least one such intersection must be empty). In that case, for a Riemannian manifold, you can get not just connectedness, but contractibility for all (nonempty) finite intersections of basis elements, by taking a neighborhood basis to consist of geodesically convex neighborhoods. (An open set $U$ is geodesically convex if any two points in $U$ are connected by a unique geodesic in $U$, one whose length is the distance between the points). For it is immediate that any (nonempty) intersection of geodesically convex neighborhoods is also geodesically convex, hence contractible. Details are spelled out in this nLab article.