# Show that the same is not true in $\mathbb{ R}^ n$, if $n \geq 2$. [duplicate]

We know that any open $$(\mathbb{ R}, d)$$ is the union of at most countable open balls disjunct two by two. Show that the same is not true in $$\mathbb{ R}^ n$$, if $$n \geq 2$$.

I'm trying to show a counter example where I display an open set of R where A is not a joint of disjoint open balls.

• The normal English for your disjunct two by two is pairwise disjoint. Feb 14, 2021 at 23:02
• You can use any open rectangle, together with the fact that the union of two or more pairwise disjoint open balls is not connected. Feb 14, 2021 at 23:04
• I thought to show that there are no open countable and disjunct balls in pairs whose union is the open square $(0,1)^2\subset \mathbb{R}^2$. I know it's because of the singing, but I don't know how to write it in mathematical terms.
– User
Feb 14, 2021 at 23:11
• Use the fact that the open unit square is connected, and prove that if $\mathscr{U}$ is a countable family of pairwise disjoint open balls, then $\bigcup\mathscr{U}$ is not connected. (In fact its components are the members of $\mathscr{U}$.) Feb 14, 2021 at 23:15