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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. enter image description here

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.

EEEB
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  • How do you want it to intersect exactly? Because why not take the whole manifold? – Randall Feb 25 '19 at 11:48
  • Because it's a manifold, I can't parametrize all of it by a single chart. I am interested in describing the curve in the charts. – EEEB Feb 25 '19 at 11:52
  • Your question is phrased in open sets. Maybe it should be rephrased in charts. – Randall Feb 25 '19 at 11:53
  • Is this an important distinction? Aren't charts defined on open sets? Besides; in the definition of the manifold, each open set is homeomorphic to some open set in $\mathbb{R}^2$. In other words, to each open set there always corresponds a chart. – EEEB Feb 25 '19 at 11:55

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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.

ARA
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