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Compact Space: Simple Definition, Examples

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While there are a couple of definitions of compact space, perhaps the easiest way is to understand the concept on the number line. A bounded, closed interval on the real number line is compact; open intervals are not compact.

compact space

The closed interval [2, 3] (green) is an example of a compact space. The other two intervals are open and so are not compact.



While “compact” might lead you to think “small” in size, this isn’t true in general [1]; Many compact spaces are very large. Basically, compact spaces have stopping points, where you can go no further (think of the closed bracket as a barrier stopping you at 0 on the left and 1 on the right). With non-compact spaces, you can move on until infinity; let’s say you tried to get close to 1 on the open interval (0, 1). You could get within a tenth (0.9) a hundredth (0.99), a thousandth (.999) and so on ad infinitum.

The more precise definition of compactness is not intuitive to grasp: a space is compact if every open cover has a finite subcover. Open covers are collections of open sets that covers a space; in other words, all points of a space is in another member of the collection. Finite subcover refers to the fact that a finite number of these sets can be chosen to also cover the space. Fortunately, you don’t need to dive deep into the depths of topology for most calculus purposes; it’s usually enough to understand that if a space is bounded and closed, then it is compact.

Why is a Compact Spaces Useful?

A compact space is mostly used in the study of functions defined on those spaces. Compactness tends to make a function “well behaved” for analysis.

A compact space acts like a finite space, which allows for making easier proofs. For example, you can often find a minima and maxima in a compact space; on the other hand, is a space is non-compact you have to find suprema and infina instead [2]. Another handy property is that you can define the infinite number line (i.e., the set of all rea numbers) with a finite number of intervals.

Another benefit of compact spaces, and perhaps the most important one in calculus, is that integrals of real-valued, continuous functions on a closed interval [a, b] are always defined because [a, b] is compact.


References

[1] Compactness. Retrieved November 3, 2021 from: https://www.msc.uky.edu/droyster/courses/fall99/math4181/classnotes/notes5.pdf
[2] Compactness. Retrieved November 3, 2021 from: math.toronto.edu/ivan/mat327/docs/notes/16-compact.pdf

CITE THIS AS:
Stephanie Glen. "Compact Space: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/compact-space/
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