# Consequences of Compactness in Physics

If we understand spacetime as a $4$-dimensional manifold $M$, from the point of view of physics what are the consquences of a subset of it being compact? My point here is simple: in math we usually think of compactness as some analogue of finiteness because it shares many properties with finite sets, but what are the consequences of this when we deal with physics?

Of course, we need not to go into relativity, we can even think about the usual three space $\mathbb{R}^3$. What are again the consequences of a set $A \subset \mathbb{R}^3$ being compact? Are there any cool things we can get out from this, or we simply use compactness in physics to grant the mathematical properties desired without having any direct impact in the way we understand and interpret those sets?

Thanks very much in advance.

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For a space to be compact, doesn't it need to be finite and bounded? Thus it seems like @PeterKravchuk is right - this would only be interesting to consider for certain functions... –  zhermes May 6 '13 at 20:30
@zhermes, Oops, I have deleted that comment, and posted a version as an answer. –  Peter Kravchuk May 6 '13 at 20:36
–  Qmechanic May 6 '13 at 21:00
Do you really mean, what interesting properties can be derived from a compact subset of space-time or what physical consequences a compact space-time (e.g compactifed Minkowsi) would imply? –  Tobias Diez May 7 '13 at 15:42
Yes @altertoby, it was exactly that what I meant. –  user1620696 May 7 '13 at 16:09

## 1 Answer

I think that you should reformulate your question. $A$ can be a single point as well as a closed ball, and I can't see what good physical insights you can make into common properties of these examples. I think that usually we need compactness to think about maxima and minima of continious functions, and boundedness of such function, but it is a pretty mathematical viewpoint.

In fact, we usually deal with very good sets. Like, they have piecewise-smooth boundary etc, which form only a tiny part of all compact sets. Compact sets are just closed bounded sets, and compactness allows really crazy sets. For example, the Сantor set is compact, as well as many other sets that we almost never encounter in Physics.

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