0
$\begingroup$

I'm trying to prove that interval $(0,1]$ is not compact by showing it doesn't have Heine-borel property.

I know a set is compact if a set is closed and bounded or has BW property or has Heine-borel property. But I'm trying to use heine-borel property to prove that it is not compact. I know I have to use the definition of open cover to prove this, but I don't know how to begin.

my guess: in order to prove $(0,1]$ is not compact by showing it doesn't have heine-borel property, is to show that there exists open cover $(0,1]$ that cannot be reduced to a finite subcover. but then what would be $\mathscr{U}$?

$\endgroup$
  • $\begingroup$ take the obvious open cover: the collection $(1/n,2)$, say, for positive integers $n$. $\endgroup$ – symplectomorphic Oct 24 '16 at 3:18
  • $\begingroup$ sorry i don't understand open cover too well. how is $(1/n,2)$ an obvious open cover? @symplectomorphic $\endgroup$ – Allie Oct 24 '16 at 3:19
  • $\begingroup$ sorry, what is the Heine Borel property? $\endgroup$ – Jorge Fernández Hidalgo Oct 24 '16 at 3:20
  • $\begingroup$ state the definition of an open cover and think about my example. $\endgroup$ – symplectomorphic Oct 24 '16 at 3:21
  • $\begingroup$ @symplectomorphic ah i was dumb. i can even make $(1/2^n,2)$ $\endgroup$ – Allie Oct 24 '16 at 3:32
1
$\begingroup$

You want to construct an open cover for which no finite subcover can still cover all of the interval $(0,1]$. One way you might do this is to take a collection of covers $U_n = (a_n, 2)$, where $a_n \to 0$.

If you create $a_n$ such that all $a_n > 0$, then clearly no finite subcover will still cover the interval.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.