I am reading a proof about partition of unity in Pollack Guillemin and he introduced this set in the title.
So let the set $X \subset \mathbb{R}^n$ and denote any covering of $X$ by $(U_\alpha)$. The sets $U_\alpha$ is open in $X$ so say $U_\alpha = X \cap W_\alpha.$ They set $W = \cup_\alpha W_\alpha$ and let $(K_j)$ be a nested sequence of compact sets that exhaust the set $W$, $\cup_j K_j = W$ and $K_{j} \subset int (K_{j+1})$
Then they write "for instance" $K_i = \{x \in X: |x| < i, d(x,\mathbb{R}^n-X) \geq 1/i \}.$ How do I interpret these sets? I can't figure out what they are suppose to look like. I know what $|x| < i$ means, it means $B_i(0).$ (thanks for correction)
I cannot visualize what combining $|x|<i$ with $d(x,\mathbb{R}^n-X) \geq 1/i$ are doing, are these also points that can get closer points outside the set $X$?
Let $(M,d)$ be a metric space, $x\in M$ and $A\subseteq M$. The distance from $x$ to $A$ is defined as $$d(x,A)=inf\{d(x,y):y\in A\}$$ Returning to your case, $d(x,\mathbb{R}^n\setminus X)\geq\frac{1}{i}$ means that for each $y\notin X$, $d(x,y)\geq \frac{1}{i}$. This can be seen as isolating each $x\in X$ with a ball of radius $\frac{1}{i}$.
Note: In paragraph 3 you say that you read the first condition as $B_i(x)$, but it is actually $B_i(0)$, maybe it was a typo, but is better to be sure.
Here is a picture i drawed of the situation, in it there is a part of $X$ which is a distance $1/i$ from the border of $X$.