Classify each of the following sets as open, closed, or neither \begin{align*}
& 1) \quad\{x : |x − 5| < 1\} \\
& 2) \quad \{x : (x − 3)^2 ≥ 1\}\\
& 3) \quad \{x : 1 ≤ (x − 4)^2 < 4\}
\end{align*}
my answers I have so far:
$1)$ I solved the inequality and it worked out to be the set $(4,6)$, so I am assuming that this is open.
$2)$ I solved the inequality and it worked out to be $[2,4]$, so I am assuming that this is closed.
$3)$ Not sure how to work this one out given that one side is greater than or equal and the other is just less than.
 A: You're right that $(1)$ is open and $(2)$ is closed. To work out $(3)$, first work out the inequality $1\leq (x-4)^{2}$ and then work out the inequality $(x-4)^{2}<4$. Since our set satisfies both of these conditions, then you must consider the intersection of the two outcomes you obtained. Finally, recall that intervals of the form $[a,b)$ and $(a,b]$ are neither open or closed.
A: An open set means every point is an interior point.  An interior point is one in which every neighborhood is a subset of the set.
What are the interior points of (1,2)? Is every point an interior point?  (Hint: yes)  Is the set open.
What are the interior points of [1,2]?  Is every point an interior point?  Which ones aren't?  Why not?  Is it open?  What about [1,2)?
A closed set means every limit point of the set is an element of a set.
What are the limit points of (1,2)? of [1, 2)? of [1,2]?  (Hint:  all three of those sets have the same limit points.  Which of the limit points are members of which sets and which limit points are not members of which sets?  Which sets are closed?
Which of (1,2), [1,2] and [1,2) are open? which are closed? which are both? (Hint: none) and which are neither?
