# 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.

• Sorry my textbook states that an open set contains none of its boundary points. I guess it means that both sides. And one more question to ask. Is every finite set is closed then?? By applying same statement that the complement of open set is closed?? Thanks. Feb 21, 2015 at 0:08
• @eChung00 you are correct; I'm sorry I had in my mind that a boundary point was the same as limit point. Feb 21, 2015 at 0:12
• Thank you! It is more clear now!! Your explanation is very good.!! Thanks Feb 21, 2015 at 0:30
• No problem, although I used some pretty big theorems in my answer. You should definitely try to prove that: (1) A set is closed iff it's complement is open (2) Any finite intersection of closed sets is closed and (3) An arbitrary union of open sets is open. These three should help clarify all these definitions for you, and you will use them frequently as you go on in math. Feb 21, 2015 at 0:34
• Thanks for the advice. I will definitely try them! ^.^ Feb 21, 2015 at 0:41

A very quick way to guess open and close set can be simply imagining the existence of neighborhood of a point. Set is open, if it contains all its interior point. A point is called interior point if the neighborhood of the point is contained inside the set. Take an example (4,7), you may bring any point 'p' between 4 and 7 (not including them) you can have N(p) contained in (4,7), So any point of (4,7) is an interior point. Since (4,7) contains all its limit point so (4,7) is open. Another example say integers, you may take any arbitrary integer 'z', but N(z) is not contained in integers. For e.g say z=4, if we take epsilon =1/2, then the N(4)= (4-1/2, 4+1/2) which is clearly not contained in integers. A set is said to be closed if it contains its derived set, I.e- the set of all limit points. Points are said to be limit points if the neighborhood of the point contains infinitely many number of points of that set. For e.g. - [4,7], here any real number between 4 and 7 (including both) are limit points. Because if we take any point 'r' between [4,7], then the intersection of N(r) and [4,7] has infinitely many points. But, if we consider integers, let's consider again 4 with epsilon =1/2, then N(4)= (4-1/2,4+1/2) intersection integers hasn't infinitely many points. Hence 4 is not a limit point of integers. Similarly we may take any integers we will find that the set of integers has no limit point. Since {} (empty set) is a aubset of any set, so from this angle we consider integers to be closed as it contains the set of all its limit points that is {}.

But let's go with it. For any finite set $\{x_n\}$, we can calculate all the distances and take the minimum (there's a finite number), let's call it $\mu$. Taking $\epsilon < \mu$ in the definition of the limitof a sequence, we realise that all convergent sequence are eventually constant. As boundary points are exactly those points that are the limits of (sub)sequences, we realise that for any finite set $A$, the set of boundary points is also $A$, thus we have that any finite set is closed.
Finite sets are closed (in Hausdorff spaces like $\mathbb{R}$). An easy way to see this perhaps, is by noting that the complement of a set containing just one point $x$, is $(-\infty,x)\cup (x,\infty)$ which is the union of two opens and hence open. Since the complement of an open is closed (in topology this is actually the definition of closed sets), $\{x\}$ is closed. Now use that finite unions of closed sets are closed.