Let $(X,d)$ be a connected compact metric space. Does there exist a finite family $B_1,...,B_n$ of open balls in $X$ with a given radius $R>0$ such that $B_i \cap B_{i+1} \neq \emptyset$ for $i=1,\ldots ,n-1$ and $X=\bigcup_{k=1}^n B_i$.
Thanks.
Let $(X,d)$ be a connected compact metric space. Does there exist a finite family $B_1,...,B_n$ of open balls in $X$ with a given radius $R>0$ such that $B_i \cap B_{i+1} \neq \emptyset$ for $i=1,\ldots ,n-1$ and $X=\bigcup_{k=1}^n B_i$.
Thanks.
Let ${\cal B}=\{B_1,B_2,\ldots, B_n\}$ be an open cover of $X$ of balls of radius $R$.
Claim: Let $x$ and $y$ be in $X$. Then there is a "chain" from $x$ to $y$ consisting of elements of $\cal B$. That is, there is a sequence of sets $B_{n_1},B_{n_2},\ldots, B_{n_k}$ from $\cal B$ with $x\in B_{n_1}$, $y\in B_{n_k}$ and $B_{n_i}\cap B_{n_{i+1}}\ne\emptyset$ for each admissable $i$.
To see why the claim is true, you can use the argument presented in the second link mentioned by Martin Sleziak in the comments.
So, between any two points $x$ and $y$ in $X$, there is a chain from $x$ to $y$ consisting of open balls of radius $R$. Now let $x_1$, $x_2$, $\ldots\,$, $x_n$, be the centers of the open balls in $\cal B$. One obtains the desired finite family of open balls by constructing chains from $x_1$ to $x_2$, then from $x_2$ to $x_3$, and so on (note that this usually won't give the smallest collection).