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.
[Math] How to determine if a set is open or closed?
real-analysis
Related Solutions
Let $X$ be a compact Hausdorff space and $p\in X$ apoint such that $Y:=X-\{p\}$ is not closed. For each $x\in Y$ there exist disjoint open sets $x\in U_x,p\in V_x$. Then the $U_x$ cover $Y$. Assume there is a finite subcover $U_{x_1}\cup\ldots\cup U_{x_n}$. Then this subcover misses the open set $V_{x_1}\cap \ldots\cap V_{x_n}$ which contains $p$ ans must be strictly larger that $\{p\}$ because $\{p\}$ is not open in $X$.
Actually, if you haven't already learned this, you will learn that in $\Bbb R^{n}$, a set is compact if any only if it is both closed and bounded.
The set $(0,1)$ is bounded, but it is not closed, so it can't be compact.
The solution to your exercise of finding an open cover with no finite subcover proves that $(0,1)$ is not compact, because the definition of a set being compact is that every open cover of the set has a finite subcover.
So, the whole trick to finding an open cover with no finite subcover is this:
$(0,1)$ is not closed, and so it doesn't contain all of its limit points (it's easy to see that the two it doesn't contain are $0$ and $1$). Well, can you construct an open cover whose open sets get closer and closer and closer to at least one (or maybe both) of these end points, but never quite reaches them? Hint: since we are in $\Bbb R$, think about how you would get closer and closer to a real number $r$ without ever actually being $r$. If you want to get closer from above, then $\{r + \frac{1}{n} \}_{n = 1}^{\infty}$ is a sequence that gets closer to $r$ from above. What would be a sequence that gets close to $r$ from below? You should hopefully say $\{r - \frac{1}{n} \}_{n = 1}^{\infty}$.
Anyway, so maybe we can construct our open intervals so that the end points get closer and closer to the limit points $0$ and $1$, but finitely many of the sets would always have gaps between them and $0$ and $1$... hmm...
Well, we could do $(0 + \frac{1}{3}, 1 - \frac{1}{3}) \cup (0 + \frac{1}{4}, 1 - \frac{1}{4}) (0 + \frac{1}{5}, 1 - \frac{1}{5}) \cup ...$.
These open sets are in $(0,1)$ and have end points getting closer and closer to $0$ and $1$, but any finite number of them won't contain all of $(0,1)$ (why?). If we use a formula to define these open intervals, it would be $\{ (0 + \frac{1}{n}, 1 - \frac{1}{n}) \}_{n = 3}^{\infty}$.
Best Answer
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.