Understanding Compact Sets and Their Role in Continuity

[darkSun]

I'm taking real analysis and struggling a bit. In class today our professor was saying something about how a function may not be continuous on a non compact set or something, but anyway, he drew the closed interval from 0 to 1 but looped one end back to the middle of the interval.

__
| |
----------

Kind of like this, so that an end point and some point in the middle coincide. So my question is, why isn't the looped interval [a,b] compact? Wouldn't it be compact for the same reason [a,b], that is, wouldn't it have the same finite subcover?

 

[Office_Shredder]

I think this is what he was referring to:

Let's say the loop brings the point b to the midpoint (a+b)/2. Then if you have a cover of [a,b) you covered b also since it's the same point as (a+b)/2. But [a,b) can have bad covers. For example with the interval [0,1] with 1 looped around to 1/2 the sets [0,1-1/n) are open and cover [0,1] in this case, but there is no finite subcover.

This is a subtle topological point; in the normal induced topology from R² those are not open sets, but the point is you can define a topology on the loop that makes it non-compact.

 

[darkSun]

Oh I understand! It's basically the same interval except an endpoint or a point in the middle is removed, and I can see why that would be non-compact. Thank you, Office_Shredder.

And this is kinda unrelated, but the definition of compactness using open covers and finite subcovers seems really weird to me. And it seems very difficult to use this definition when proving whether or not metric spaces are compact. Is this definition used like this often? Seems very clunky to me.

 

[lurflurf]

It is cluncky, but at least it gets to what compactness is about. Compactness is the next best thing to finiteness.

 

[HallsofIvy]

I think I've heard that before!

 

[snipez90]

And this is kinda unrelated, but the definition of compactness using open covers and finite subcovers seems really weird to me. And it seems very difficult to use this definition when proving whether or not metric spaces are compact. Is this definition used like this often? Seems very clunky to me.

Um, I actually find the definition fairly intuitive/visual? My analysis teacher introduced the concept today, and this is really the first time it's been mentioned in the course. I've studied sequential compactness before, so I can see how the open cover definition might be harder to use. Or rather, I would say if you're not familiar with proving the various topological theorems (of course with the exception of compactness theorems... the concept is usually introduced after the basics have been covered) in the context of metric spaces, then compactness proofs may seem weird.

But once you are comfortable with working with open balls and such, theorems such as compactness implies closed and bounded and say, a closed subset of a compact set is compact can be gotten just by drawing out the geometric situation and thinking (basically what our teacher did for these two proofs, the third was about the continuous image of a compact set being compact, which is also much cleaner than the proof for the corresponding theorem for sequential compactness).

 

[darkSun]

It would help if our teacher made it more intuitive w/ geometry or something... I've had no exposure to topology so this is very new to me. But ill try to think of it like you said, and as an analogue of continuous sets. Thanks