I read from Number of components of intersection of coordinate balls. that given arbitrary second countable basis $\mathcal{B}$ of a topological manifold $M$ consisting open subsets homeomorphic to open balls, the intersection of any two of them may not be path connected.
However, is there some well choosed basis such that the intersections do be path connected, or even simply connected? I think the answer for dimension one and two should be true for the classification of such topological manifolds. Nevertheless I cannot figure out if it's positive or not for the general cases.