Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
Given: 1) is (-2,4) compact by applying Heine-Borel Property? and 2) is [-2,4] compact by applying Heine-Borel property?
I've followed the proof of Heine-Borel property and i have attempted to apply the proof to answer the following and i am not quite sure how to prove? it...
i know that [a,b], every open cover has a finite subcover and a set of real numbers is compact if and only if it is closed and bounded.
So for 1) no and 2) yes?
If you know Heine-Borel Theorem, then its fairly obvious wich is compact and wich isnt, $(-2,4)$ is not closed so it cannot be compact, and $[-2,4]$ is both closed and bounded, so it is compact. If you want to do it more 'hands on' you can try the following:
For $1)$: Try to construct a sequence wich doesnt have any subsequence converging in the space $(-2,4)$
For $2)$ How do you know $[a,b]$ is compact? Hint: Let $I_n$ be an open cover of $[a,b]$, then take
$$C = \{x : \text{ such that $[a,x]$ has a finite subcover} \}$$
Prove $\sup C$ exists and that $\sup C = \max C = b$
This will yield a general result for $[-2,4]$ you can even replace $a = -2, b = 4$ and use the same hint.
