How do I determine if a set is open or closed?? I have a question about open and closed sets. As far as I know, a open set is a set that do not contains its boundary points. A closed set is a set that contains its boundary points. If we think of an interval on real line, such as $(0,1)$ and $[0,1]$, the first interval is open and the second one is closed. However, If I am given finite set such as $\{1 ,2 ,3 \}$ or $\{10, 19, -10\}$ in $\mathbb{R}$, how do I determine if the set is open or closed?? From those finite sets, how do I know what is its boundary points?? I am having real analysis class and having hard time. Can anyone give some explanation with example?? Thanks.
First and foremost, it is important to know that open and closed are not opposites; i.e, a set that is not closed is not necessarily open. Sometimes sets can be neither open nor closed. For example, $[0,1)$. Sometimes sets can be both open and closed. For example, the emptyset or $\Bbb{R}$. One way to define an open set on the real number line is as follows:

$S \subset \Bbb{R}$ is open iff for all $s \in S$, there exists an interval of the form $(a,b)$ such that $s\in(a,b) \subset S$.  

Another way to tell if a set is open is if it is the complement of a closed set. If $C$ is a closed set, then $\Bbb{R} \setminus C$ is open. Let's consider the union of open sets $(-\infty,1)\cup(1,2)\cup(2,3)\cup(3,\infty)$. This union is open (although you should prove that any union of open sets is open so you can know this). Now, the complement is $$\Bbb{R} \setminus [(-\infty,1)\cup(1,2)\cup(2,3)\cup(3,\infty)]= \{1,2,3 \}$$ so we now see that the complement of $\{1,2,3 \}$ is open, allowing us to deduce that $\{1,2,3 \}$ is closed. Read the definitions carefully of open sets, closed sets, limit points and boundary points. A clear understanding of the differences and how they interact will take you far in real analysis and topology.
A finite set is equal to its set of boundary points.