An example of an infinite open cover of the interval (0,1) that has no finite subcover

Question

I've been having a hard time solving this problem that I was given in class. The problem states " Give an example of an infinite open cover of the interval (0,1) that has no finite subcover."

I know that the set has to be compact and that both 0 and 1 are limit points. Aside from those two known factors I'm at a complete loss as to how to go about solving this problem.

Answer 1

Think about... $$ \bigcup_{n=1}^{\infty}\left(0+\frac{1}{n},1\right) $$

Answer 2

You can take the open cover $\{(0+\frac{1}{n}, 1-\frac{1}{n})\mid n>2\} $. It has no finite subcover because if it does simply take the largest $n$ and $\frac{1}{2n}$ is not included.

Answer 3

Since you have the real analysis tag, I'm assuming you are using the standard topology on $\Bbb{R}$, so open sets look like intervals of the form $(a,b)$. An open cover of $(0,1)$ could look something like $$\bigcup_{i=1}^\infty \left(0,1-\frac{1}{i} \right)$$ But you need to verify this is an open cover. That is, show $$(0,1) \subseteq \bigcup_{i=1}^\infty \left(0,1-\frac{1}{i} \right)$$ and further show that if $K \subset \Bbb{N}$ is any finite subset of $\Bbb{N}$ that $$(0,1) \not\subseteq \bigcup_{i\in K} \left(0,1-\frac{1}{i} \right)$$ which demonstrates that the open cover has no finite subcover.

Answer 4

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 $(0 + \frac{1}{3}, 1 - \frac{1}{3}) \cup (0 + \frac{1}{4}, 1 - \frac{1}{4}) \cup (0 + \frac{1}{5}, 1 - \frac{1}{5}) \cup \ldots$.

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 $\{ (0 + \frac{1}{n}, 1 - \frac{1}{n}) \}_{n = 3}^{\infty}$.