Identify the compact subsets of R

[jeckt]

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!

 

[micromass]

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?

 

[jeckt]

• ]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.

 

[micromass]

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...

 

[jeckt]

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].

 

[micromass]

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...

 

[jeckt]

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!