In the Wikipedia's Page is wrote:
Exhaustion by compact sets of an open set $U$ in the Euclidean space $\mathbb{R}^n$ is an increasing sequence of compact sets ${\displaystyle K_{j}}\subseteq U$, where by increasing we mean ${\displaystyle K_{j}}\subseteq{\displaystyle K_{j+1}}$, with the limit (union) of the sequence being $U$ (i.e. $U=\bigcup_{j=1}^\infty K_j$).
Sometimes one requires the sequence of compact sets to satisfy one more property—that ${\displaystyle K_{j}}$ is contained in the interior of ${\displaystyle K_{j+1}}$ for each ${\displaystyle j}$. This, however, is dispensed in $\mathbb{R}^n$.
My question:
I am confused and I do not understand why in $\mathbb{R}^n$, ${\displaystyle K_{j}}$ is contained in the interior of ${\displaystyle K_{j+1}}$ is dispensed. Is it right? Why?
Here is an example to think about. Let $n=1$. Consider the family of compact subsets $K_i\subset {\mathbb R}$ given by $$ K_i=[-i, - i^{-1}]\cup \{0\} \cup [i^{-1}, i]. $$ This family forms an exhaustion of ${\mathbb R}$ by compact subsets (in the sense of the Wikipedia article), such that for all $i$, $j$, $K_i$ is not contained in the interior of $K_j$.