A friend of mine asked me if the following question is true:     Question: Let $M$ be a compact connected smooth manifold without boundary, then if $a_1,$ $a_2$ $\in$ $M$, there is a chart $(\varphi,U)$ of $M$ such that $a_1,$ $a_2$ $\in$ $U$.    Despite this seems like a lie, I wasn't able to find a counterexample or prove this statement.  Anyone know how to demonstrate this or know a counterexample for this question?
Some say that we can connect two points with a continuous curve and a small contractible neighborhood of the path together with the charts of the points can be regarded as the chart. But I don't know how to verify this statement.
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