Given a smooth manifold, say a two-sphere, and given two disjoint open sets on it, containing two different points of the manifold (Hausdorff property) can I find a third open set that intersects them both?
If yes; in what property of the manifold does this rely on? My motivation for asking is because I imagine a curve on the manifold with initial and final points being $p$ and $q$, and I want to understand if I can describe the entirety of the curve in the charts. 
p.s.: Thinking about it maybe the compactness of the manifold is required. If the manifold is compact, and if $M=\cup_i U_i$, for even countably infinite open sets, then they must intersect, otherwise the volume would be infinite. But still, I am uncertain if this implies what I am asking: namely; given any two disjoint open sets, that there exists a third intersecting them both.
If we are given an open cover $ \{ U_i \} $ of domains of charts of $M$, any path $ \gamma:[0,1]\rightarrow M $ is the composite of finitely many paths $ \gamma_i $, each of which does lie in a single $ U_i $, because $ \gamma([0,1]) $ is compact.