Is the interior of the union of $n$ closed balls equal to the union of the interiors of the $n$ closed balls? I am reading "Calculus on Manifolds" by Michael Spivak.
I am solving problem 1-22 on p.10 now.
If the following equality holds, I can solve the problem.
Let $B_1, \dots, B_n$ be closed balls in $\mathbb{R}^m$.
Intuitively, I guess the following equality holds, but I cannot prove that.
Does the following equality hold?
$$\operatorname{Int}(B_1 \cup \dots \cup B_n) = \operatorname{Int}(B_1) \cup \dots \cup \operatorname{Int}(B_n)$$

I think your intuition is correct in one sens
this a counterexample take two intervals $[1,2]$and $[2,3]$
it's clear that the interior of the union is not equal to the union of the interior  even if your sets are closed
