Identify the compact subsets of R

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Homework Statement



Identify the compact subsets of \mathbb{R} with topology \tau:= \{ \emptyset , \mathbb{R}\} \cup \{ (-\infty , \alpha) | \alpha \in \mathbb{R}\}.

just need help on how would you actually go about finding it. I usually just find it by thinking about it.

The Attempt at a Solution



  • \emptyset
  • [a,b] with a,b\in \mathbb{R}
  • \{x\} with x\in \mathbb{R}

I was also thinking about subset with only two points e.g. \{ x,y\} with x,y\in \mathbb{R}. They are compact but then...if i can keep doing that i'll get a countable infinite set, hmmm which i think should also be compact.

thanks!
 
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Beautiful question!

Anyways, let me first comment on the work you've already done. You've deduced that [a,b] is compact, but also ]a,b] is compact, is it?
Also {x} is compact, but finite unions of compact sets are also compact! Thus finite sets will also be compact.
Countable sets are not necessarily compact, for example, take \{-1/n~\vert~n>0\}, this will not be compact (since we didn't include it's limit point)

Of course, there will be a lot of compact sets, so listing them all will not be possible. What they want from you is some kind of criterion to easily decide whether a set is compact or not.

Here are two things to think about:
1) Take a set S, and cover it with \{]-\infty,s[~\vert~s\in S\}. When can we find a finite subcover of this?

2) What is the difference between compact sets ]a,b], [a,b], {x}, \{1/n~\vert~n>0\}, and non-compact sets [a,b[, \{-1/n~\vert~n>0\}. Can you find what is different about these sets?
 
micromass said:
  • ]a,b]
  • Countable sets are not necessarily compact, for example, take \{-1/n~\vert~n>0\}, this will not be compact (since we didn't include it's limit point)

What is ]a,b]? notation, I've never seen it before.

As for the second point, wouldn't a subcover such as (-\infty , a) where a is any number greater than 0.

Sorry about this -I'm really slow when it comes to maths.
 
jeckt said:
What is ]a,b]? notation, I've never seen it before.

It's the same as (a,b].

As for the second point, wouldn't a subcover such as (-\infty , a) where a is any number greater than 0.

Of course, ]-\infty,a[ will cover the set. But compactness doesn't mean that you can find a finite cover. Compactness means that for any cover, you can find a finite subcover. So, take

\{]-\infty,-1/n[~\vert~n>0\}

this will be a cover of \{-1/n~\vert~n>0\} but without a finite subcover...
 
Thanks micromass! As for your original questions...I'll have a think about it - but from a quick glance, it seems that the -\infty plays a large roll in being able to have compact sets of the form (a,b].
 
jeckt said:
Thanks micromass! As for your original questions...I'll have a think about it - but from a quick glance, it seems that the -\infty plays a large roll in being able to have compact sets of the form (a,b].

That is indeed correct! The consequence about having -\infty is that we can do anything we want on the left side. Only the right side of sets will matter. If you see what I mean...
 
Yeah I totally get you micromass! been so busy - haven't been able to reply. So gathering all that together the compact subsets are thus:

  • \emptyset
  • For a_{k},b_{k}\in \mathbb{R} \mbox{ s.t. } a_{k}<b_{k}\ \forall k \mbox{ and } n\in \mathbb{Z}^{+} \mbox{ s.t } n<\infty
    • \bigcup\limits_{k=1}^{n} \{ a_{k}\}
    • \bigcup\limits_{k=1}^{n} ( a_{k},b_{k}]
    • \bigcup\limits_{k=1}^{n} [ a_{k},b_{k}]

\mbox{Thus any subset of } \mathbb{R} \mbox{ that has a maximal element or is the empty set. This is indeed an interesting question.}

and again thanks micromass!
 
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