Showing that $[0,1]$ is compact

Let's choose an open covering for $\left [ 0,1 \right ]$. For example $$\left \{ \left ( \frac 1 n,1-\frac 1 n \right ) \mid n\in \{ 3,4,\dots\} \right \}.$$ How can one choose a finite open subcover to prove compactness?

Solution 1:

Here is a quick and elegant proof of the actual result that $[0,1]$ is compact based on real induction:

Let $\mathcal{O}$ be an (arbitrary!) open cover. Let $P$ be the set of points $x$ in $[0,1]$ such that $[0,x]$ can be covered by finitely many elements of $\mathcal{O}$. We have $0\in P$ and $P$ is bounded above by $1$. Therefore, $P$ has a supremum $s$.

We first show that $[0,s]$ can be covered by finitely many sets in $\mathcal{O}$. This is trivial when $s=0$, so assume $s>0$. Let $O_s\in\mathcal{O}$ be a set containing $s$. Then there is an $\epsilon \in (0, s)$ such that $(s-\epsilon,s]\subseteq O_s$. By assumption, there is a finite subcover of $[0,s-\epsilon/2]$. By adding $O_s$ to that finite subcovering, we get a finite subcovering of $[0,s]$.

We now show that $s=1$. Suppose $s<1$ and let $O_s\in\mathcal{O}$ be a set containing $s$. Then there is an $\epsilon>0$ such that $[s,s+\epsilon)\subseteq O_s$. So taking a finite subcover of $[0,s]$ and adding the set $O_s$ gives us a finite subcover of $[0,s+\epsilon/2]$, contradicting the construction of $s$.

Solution 2:

The collection of sets above is not an open cover of $[0,1]$. The points $0$ and the point $1$ are not in any of them.

Also you can not prove compactness of anything by taking an open cover and finding a subcover.

The definition of compactness is that for all open covers, there exists a finite subcover.

If you want to prove compactness for the interval $[0,1]$, one way is to use the Heine-Borel Theorem that asserts that compact subsets of $\mathbb{R}$ are exactly those closed and bounded subsets.

Solution 3:

Answering your question would not prove compactness. The condition for compactness is that every open cover has a finite subcover, not just that a single given open cover has a finite subcover. A proof that $[0,1]$ is compact with respect to the Euclidean metric topology can be found here.

Solution 4:

I assume that you would like to prove that it is compact with respect to the usual topology. Since that topology is "generated" by a metric the topological compactness is equvalent with metrical compactness, i.e. the condition stating that every sequence has converging subsequence.

Observe that every sequence of elements of the interval $[0,1]$ is bounded. Therefore, by the Bolzano-Weierstrass theorem, it has a convergent subsequence.

Notice that the Bolzano-Weierstrass theorem can be proved by analytical arguments, so we are not in the vicious circle.