ε environment of number 1

Question

I am stuck with a problem it asks me to find the $\epsilon$ environment of the number $X=1$ and find for which $\epsilon$. I got a few intervals to solve for, here are the intervals: $(0,2) , (\frac{2}{3},\frac{4}{3}), [-1,3)$

I have no idea which $\epsilon$ environment am I allowed to choose, for the first interval $(0,2)$ is it correct to write $(X-\epsilon,X+\epsilon) ?$ I don't understand the problem how am I suppose to find the $\epsilon$ environment without using limits ?

Answer 1

"$\epsilon$ environment of $X$" means the open interval from $X-\epsilon$ to $X+\epsilon$. So, if $X=1$, then $(0,2)$ is the $\epsilon$ environment of $X$ for $\epsilon=1$; $(2/3,4/3)$ is the $\epsilon$ environment of $X$ for $\epsilon=1/3$; $(-1,3)$ is the $\epsilon$ environment of $X$ for $\epsilon=2$.

I'm not sure what to make of the half-open interval $[-1,3)$ in the question.