A reasoning on compactness

Is the following reasoning correct ?

"The space $[0,1]$ is compact. Indeed, one can write:
$[0,1] = \left[0, \tfrac{2}{3}\right) \cup \left(\tfrac{1}{2}, 1\right]$ and these two intervals, $\left[0, \tfrac{2}{3}\right)$ and $\left(\tfrac{1}{2}, 1\right]$ , are open in $[0,1]$ . This shows that $[0,1]$ can be written as a finite union of open subsets."

Solutions

Reveal solutionsAre you sure? Give it a try first.

Solution by Sequoia

0 useful votes

It is true that both intervals $[0,\frac{2}{3})$ and $(\frac12,1]$ are open in $[0,1]$ , however this is not the definition of compactness.

Indeed, one has to consider an arbitrary covering of the desired set, and then deduce that there always exists a finite subcovering of our set. Which is not what we have done here since we already chose a specific covering of our set $[0,1]$ .