Confused on how to find the interior of a set I have to show whether or not the set $[0,1] ∪ [2,3]$ has an interior of $(0,1)$.
I'm still really confused how to determine the interior of a set. Can someone please explain?
 A: Draw the set $[0,1] \cup [2,3]$ on a number line.
The interior is the subset of points where for each point you can draw some open interval around it that is still contained in $[0,1] \cup [2,3]$.  In other words, the interior of $[0,1] \cup [2,3]$ is the subset of points $x$ such that for each $x$, there is some $\epsilon > 0$ with $(x - \epsilon, x + \epsilon)$ being a subset of $[0,1] \cup [2,3]$.
Can you tell from the picture below (the green shaded part is the set $[0,1] \cup [2,3]$) what the interior is?  Which points have an $\epsilon > 0$ so that $(x - \epsilon, x + \epsilon)$ is still a subset of $[0,1] \cup [2,3]$?

Hopefully, you said all points in $(0,1) \cup (2,3)$.
Why don't we include $0$ (or $1$ or $2$ or $3$)?  Observe what happens if you look at $(0 - \epsilon, 0 + \epsilon)$ for any positive number $\epsilon > 0$.  Is that interval contained in $[0,1] \cup [2,3]$?  If it is contained in this set for at least on $\epsilon$, then $0$ would be in the interior, but hopefully you see that the interval is not contained in the set.  The part $(-\epsilon, 0)$ of the interval $(0 - \epsilon, 0 + \epsilon)$ is not contained in $[0,1] \cup [2,3]$.
