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August 10

compactness

Why is compactness important? I understand the definitions and examples but am missing the applications, particularly for integration. Thanks. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D (talk) 13:40, 10 August 2020 (UTC)[reply]

Courtesy link: Compactness -> Compact space. -- ToE 15:19, 10 August 2020 (UTC)[reply]

That's kind of a broad question, but there are a lot of useful theorems (not just about integration) that require compactness somewhere. For example, the Extreme value theorem guarantees that a continuous function

f\colon X\to \mathbb {R}

attains a minimum and maximum value, but it does so under the assumption that

X

is a compact topological space. This in turn is a direct consequence of the fact that the image of a continuous function on a compact space is itself compact. Now that I think about where you're coming from in asking the question, it may depend a little. Compactness is often taught in terms of the real numbers first instead of the more general notion of topological spaces. So how to answer the question may depend a bit on that. –Deacon Vorbis (carbon • videos) 13:49, 10 August 2020 (UTC)[reply]

Yeah the extreme value theorem is close to the basic definition though. Compact sets are apparently important for measurability (this came up in another refdesk question a while back). Tightness of measures may have something to do with it. Compact support is also a term I hear a lot, but it seems like an odd concept if I'm used to smooth functions that can't totally flatline like that. But I think it comes up in probability so I wonder how. Thanks. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D (talk) 23:40, 11 August 2020 (UTC)[reply]

At a high level, compactness is useful because it allows you to turn infinite things into finite things. This often allows you to prove things more easily. Recall the definition of compactness: a space is compact if given any open cover, we can extract a finite subcover. Your open cover may be uncountable, but if the space is compact, we can instead work with a finite number of open sets just as well. There are other flavors of compactness, but passing from the infinite to the finite, where arguments are easier, is the important part.132.239.147.118 (talk) 02:00, 15 August 2020 (UTC)[reply]

Thanks, that is very helpful. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D (talk) 08:09, 15 August 2020 (UTC)[reply]

Circle group restrict to algebraic coordinates

Could we say the Circle group, intersected with the set of algebraic points on the x-y plane, is profinite?UsingNewWikiName (talk) 15:44, 10 August 2020 (UTC)[reply]

I'm not an expert with this stuff, but I don't think so. Certainly not with the usual topology since the resulting space isn't compact. Is there some motivation or extra detail behind this question? –Deacon Vorbis (carbon • videos) 17:36, 10 August 2020 (UTC)[reply]

I don't understand why that wouldn't be compact and totally disconnected under the subspace topology. Isn't the circle group compact?UsingNewWikiName (talk) 18:10, 10 August 2020 (UTC)[reply]

The circle group certainly is compact, but the subgroup of points with algebraic coordinates isn't, since it's not a closed subset (its closure is the whole circle). –Deacon Vorbis (carbon • videos) 18:16, 10 August 2020 (UTC)[reply]

You mean it's not closed under limits of sequences? Okay, but wouldn't it be compact in the sense of any open cover having a finite subcover?UsingNewWikiName (talk) 18:24, 10 August 2020 (UTC)[reply]

No. Hint: Take a sequence from your set which converges to a point not in the set and use that sequence to construct an open cover with no finite subcover. -- ToE 18:41, 10 August 2020 (UTC)[reply]

Alternately, consider why [0,1]∩{real algebraic numbers} isn't compact using either formulation and then apply that understanding to your set. -- ToE 18:51, 10 August 2020 (UTC)[reply]

Thank you. Guess I have a lot to learn.UsingNewWikiName (talk) 18:56, 10 August 2020 (UTC)[reply]

As do we all. Are you at least able to construct an example open cover of your set without a finite subcover? If not, please ask. We're not here to tutor, but this is simple enough that you shouldn't leave without it. -- ToE 20:22, 10 August 2020 (UTC)[reply]

I don't know if this is relevant, but if ρ is an algebraic element of the circle group then ρ+ρ^-1 is an algebraic real number between -2 and 2. Conversely, if r is an algebraic real number between -2 and 2 then the two solutions to ρ+ρ^-1 = r are algebraic elements of the circle group. The point is that there are more algebraic elements in the circle group than you might think. Topologically it would be two copies of [-2,2]∩{real algebraic numbers} with corresponding endpoints identified. --RDBury (talk) 22:48, 10 August 2020 (UTC)[reply]