A chart for a finite set

Question

Suppose you have a manifold $M$ and a finite set $S = \{x_1, \ldots, x_n\}$ of points in the same connected component of $M$. I am looking for criteria that ensures $S$ is contained in a connected chart.

Partial progress:

• $n = 2$: by the discussion in comments, there exists an embedded arc from $x_1$ to $x_2$. A tubular neighborhood will be the desired chart.
• $M$ compact, $x_i$'s very close: Take the lebesgue number $r$ of an atlas of $M$, with respect to the distance induced by some riemannian metric on $M$. If $d(x_1, x_i) < r$ for $i=2, \ldots, n$, then $S \subset B_r(x_1)$ which is in a chart by the lebesgue number property.

Answer 1

As it is asked, the answer to your question is trivially "yes". Just take charts $(U_i,\phi_i : U_i \to U'_i \subset \mathbb R^n)$ such that

1. $x_i \in U_i$.
2. $U_i \cap U_j = \emptyset$ for $i \ne j$.
3. $U'_i \cap U'_j = \emptyset$ for $i \ne j$.

Then $U = \bigcup U_i$ contains $S$ and the $\phi_i$ define a chart $(U, \phi : U \to U' = \bigcup U'_i)$ such that $\phi \mid_{U_i} = \phi_i$.

The set $U$ is not connected. If that is an additional requirement, more subtle arguments are needed.