Union of set and interval I'm working on finding the boundaries of sets, I feel like I understand this. However, one problem asks for the boundary of $\{1,2,3\}\cup(2,4)$ and I'm unsure as to how to take the union of an interval and a set. 
Here's my thoughts:
The union will include points $1$ and $2$ and then the interval from $2$ to $4$ but not including $4$. I'm not sure how I'd write that though. Possibly $\{1,[2,4)\}$?
 A: Let's think formally about what a boundary is. If you have a set $A$, with closure $\bar{A}$ and interior $\mathring{A}$, then the boundary of $A$ is $\partial{A} = \bar{A} \setminus \mathring{A}$.
Let $A = \{1,2,3\} \cup (2,4)$.
What is the closure of this set? The easy way is to find the points whose neighborhoods always contain some points in $A$ (the closure is the set of these points, by definition). In this case, the closure is $\{1\} \cup [2,4]$.
The interior is, by definition, the set of points who have at least one neighborhood contained totally in the set. So, the interior of this set is $(2,4)$.
The boundary of the set is therefore $\bar{A} \setminus \mathring{A} = \{1,2,4\}$, just three points!
You can always display a set in $\mathbb{R}$ as a union of points (singletons) with intervals. So, the original set would be $\{1\} \cup [2,4)$ as you essentially suggested.
A: You get the union of a bunch of points and the open set $(2,4)$, as you note, but I would write this as $\{1\}\cup[2,4)$.
