I don’t think I’ve spent more time with a mathematical definition than I did with compactness. It is an important mathematical property and one that initially left me entirely bewildered.
There are two definitions of compactness. One is the real definition, and one is a "definition" that is equivalent in some popular settings, namely the number line, the plane, and other Euclidean spaces. (The fact that the two definitions are equivalent is called the Heine-Borel theorem.)
The real definition of compactness is that a space is compact if every open cover of the space has a finite subcover. I don’t know how many times I repeated that definition to myself in my undergraduate topology class, wondering if my incantations would eventually help me understand what in the world compactness was.
Almost simultaneously, I learned the practical definition of compactness in Euclidean spaces: a set is compact if it is closed and bounded. A set is closed if it contains allpoints that are extremal in some sense; for example, a filled-in circle including the outer boundary is closed, while a filled-in circle that doesn’t include the outer boundary is not closed. Bounded is a little more like what it sounds like: points in a bounded space are all within some fixed distance of each other.
It took me a long time to connect these two ways of looking at compactness, and I’m not going to do that in this post. (If you’re taking an introduction to analysis or topology class, you might have the delightful opportunity to learn the Heine-Borel theorem for yourself. Hooray!) But I will unpack the first definition a bit. An open cover is a collection of open sets (readmore about those here) that covers a space. An example would be the set of all open intervals, which covers the real number line.
Of course, the collection of all open intervals in the number line contains a heck of a lot of intervals! Compactness asks if there is a way to whittledown that collection to a finite number of intervals and still cover the entire number line. That is, could we find a finite number of open intervals so that every point on the number line is in at least one of them? We could eliminate a lot of the intervals and still cover the line — we could, for example, only permit unit-length intervals whose endpoints were at integers or integers-and-a-half — but we could never pare our collection down to a finite number of intervals and still span the entire number line. If we reduced it to 100 unit intervals, for example, we could only cover a maximum of 100 units of length on the infinite number line, and that’s if none of the intervals overlapped! So the number line is not compact because we have found an open cover that does not have a finite subcover.
A set does not have to be infinite in length or area to be non-compact. A closed interval and an open interval make a good case study for how we can think about compactness. For convenience, we might as well look at the intervals (0,1) and [0,1]. (The first is all the real numbers between 0 and 1 not including the endpoints, the second is all the real numbers between 0 and 1 including 0 and 1.) The open interval (0,1) is not compact because we can build a covering of the interval that doesn’t have a finite subcover. We can do that by looking at all intervals of the form (1/n,1). Each one of those intervals lies within (0,1), and put together, any number in the interval (0,1) is in at least one interval of the form (1/n,1). For example, the point .0001 is in the interval (1/10001,1), even though it’s not in the intervals (1/2,1), (1/3,1), and so on up to (1/10000,1). But if we want to cover the entire interval (0,1) with only a finite subcollection, we will fail. Any finite subcollection will have a largest interval in it, whether it’s (1/10,1) or (1/10000,1) or (1/Graham’s number,1). In any case, we can find numbers between 0 and the left endpoint of the largest interval that won’t be covered by our finite subcollection.
When we add the endpoints 0 and 1, the interval becomes compact. Now the weird open cover we had no longer covers the whole interval because the points 0 and 1 aren’t any of the intervals. It’s harder to show that we couldn’t cook up a different pathological open cover, so you’ll have to take my word for it for now.
Showing that something is compact can be trickier. Proving noncompactness only requires producing one counterexample, while proving compactness requires showing that every single open cover of a space, no matter how oddly constructed, has a finite subcover. But eventually I came to a rigorous understanding of compactness and how both definitions fit together, and I lived happily ever after.
Now, years after wrestling with it for the first time, I’ve come to what Terry Tao might describe as a post-rigorous understanding of compactness. Compact means small. It is a peculiar kind of small, but at its heart, compactness is a precise way of being small in the mathematical world. The smallness is peculiar because, as in the example of the open and closed intervals (0,1) and [0,1], a set can be made “smaller” (that is, compact) by adding points to it, and it can be made “larger” (non-compact) by taking points away.
As a notion of smallness, then, compactness is a bit fraught. It’s a bit unsettling to say that a set can be “smaller” than a set that lies entirely inside it! But I think smallness is a valuable way to see compactness. A set that is compact may be large in area and complicated, but the fact that it is compact means we can interact with it in a finite way using open sets, the building blocks of topology. (For more on open sets, check out my post Change your open sets, change your life.) That’s the point of the finite subcover in the definition of compactness. That finite collection of open sets makes it possible to account for all the points in a set in a finite way. That comes up in, for example, the proof of the Heine-Borel theorem.
Before I realized compact meant small, I saw that compact sets were often easier to deal with. Continuous functions defined on compact sets have more controlled behavior than functions on non-compact sets. Compact two-dimensional surfaces have a nice classification theorem. Classifying non-compact surfaces is more difficult and less satisfying. Compact surfaces are more constrained. Non-compact ones can squirm out of your hands like blobs of rice pudding. Compact ones are more like jello: they might wobble a bit, but you can hold on to them if you don't mind getting your hands a little dirty.
The post-rigorous understanding of compactness allows the word"compact" to circlearound from something that feels like robot speak to something that aligns very closely with an English meaning of the word. I don’t know the history of the mathematical use of the word compact, so I don’t know how intentional that is. I like to think of it as a delightful accident of mathematical-linguistic convergence.
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FAQs
What is the example of compactness? ›
Definition of Compactness
For example, a finite set in any metric space (X, d) is compact. In particular, a finite subset of a discrete metric (X,d) is compact. Sequentially Compact: A metric space (X, d) is said to be sequentially compact if every sequence in X has a subsequence that converges in X.
A metric space (M, d) is said to be compact if it is both complete and totally bounded. As you might imagine, a compact space is the best of all possible worlds. Examples 8.1. (a) A subset K of ℝ is compact if and only if K is closed and bounded.
What is the importance of compactness? ›Compactness is important because many important properties of continuous functions only hold when the domain is a compact set (i.e., the function has compact support): continuous bounded. continuous there exist a , b : g ( a ) = inf g ( x ) , g ( b ) = sup g ( x ) (extreme value theorem)
What is compactness explained? ›The real definition of compactness is that a space is compact if every open cover of the space has a finite subcover.
What is a synonym for compactness? ›synonyms: concentration, denseness, density, tightness. Antonyms: dispersion, distribution.
What is a good sentence for compact? ›He was compact, probably no taller than me. He looked physically very powerful, athletic in a compact way. The Smith boy was compacting the trash. The soil settles and is compacted by the winter rain.
How do you determine compactness? ›A common compactness measure is the isoperimetric quotient, the ratio of the area of the shape to the area of a circle (the most compact shape) having the same perimeter. In the plane, this is equivalent to the Polsby–Popper test.
What is the most common definition of the word compact? ›joined or packed together; closely and firmly united; dense; solid: compact soil. arranged within a relatively small space: a compact shopping center; a compact kitchen.
What is a compact and give an example? ›compacted; compacts; compacting; compactedly
As a verb, compact means "to compress or squeeze together," like how the garbage truck compacts your bags of trash. Compact, the adjective, describes something that is tightly packed together, like your luggage that is so compact it fits in the overhead compartment.
consisting of parts that are positioned together closely or in a tidy way, using very little space: compact soil/sand.
What does compact support mean? ›
A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact support if its support is a compact set. For example, the function in its entire domain (i.e., ) does not have compact support, while any bump function does have compact support.
What are the four most commonly used measures of compactness? ›Review of suggested compactness indices resulted in the identification of four categories, based upon: 1) perimeter-area measurement, 2) single parameters of related circles, 3) direct comparison to a standard shape, and 4) dispersion of elements of a shape's area.
What is a compact answer? ›closely and firmly packed or put together; dense; solid. 2. taking little space; arranged neatly in a small space. 3. not diffuse or wordy; terse.
What does compact form mean? ›adj. 1 closely packed together; dense. 2 neatly fitted into a restricted space. 3 concise; brief. 4 well constructed; solid; firm.
What is a compact person? ›a compact person is physically small but looks strong. Synonyms and related words. Describing a person's muscles and general shape. an hourglass figure.
What is the opposite of compactness? ›| loose | airy |
|---|---|
| lax | loosened |
| slackened | insecure |
| unsecured | open |
| uncrowded | roomy |
Simple Sentences
Examples of simple sentences include the following: Joe waited for the train. The train was late. Mary and Samantha took the bus.
A simple sentence contains a subject and a verb, and it may also have an object and modifiers. However, it contains only one independent clause.
What is a sentence simple words? ›In simple terms, a sentence is a set of words that contain: a subject (what the sentence is about, the topic of the sentence), and. a predicate (what is said about the subject)
What is the compactness of an image? ›Compactness is defined as the ratio of the area of an object to the area of a circle with the same perimeter. – A circle is used as it is the object with the most compact shape.
How do you prove a set of functions is compact? ›
Furthermore, F is compact if and only if it is closed, pointwise bounded, and equicontinuous. Proof. Since C(X) is complete, a subset is complete if and only if it is closed. It follows that F is compact if and only if it is closed and totally bounded.
Does density measure compactness? ›Density is a measure of the “compactness” of matter within a substance and is defined by the equation: Density = mass/volume eq 1. The standard metric units in use for mass and volume respectively are grams and milliters or cubic centimeters.
Does compact mean big or small? ›(kəmpækt ) (kɒmpækt ) adjective [usually ADJECTIVE noun] Compact things are small or take up very little space. You use this word when you think this is a good quality.
What is a compact used for? ›Compact powders are the perfect type of powder to touch up when on the go. It is pressed into a pan and can be used to get rid of the excess oil and sweat on your face. A compact powder also adds light coverage to your skin and evens out your complexion.
What part of speech is compact? ›compact adjective (CLOSE TOGETHER)
What does compact area mean? ›Related Definitions
Compact area means all that land area actually owned or controlled by the authority by deed, lease, option, right of first refusal, or other legal or accepted instrument of land exchange.
1. Closely and firmly united or packed together; dense: compact clusters of flowers. 2. Occupying little space compared with others of its type: a compact camera; a compact car.
What is an example of functions with compact support? ›ψ(x)= {e−1/(|x|2−1),|x|<1,|x|2=∑nj=1x2j,0,|x|≥1, can serve as an example of an infinitely-differentiable function of compact support in a domain Ω containing the sphere |x|≤1.
What does compact closure mean? ›A set has compact closure if its set closure is compact. Typically, compact closure is equivalent to the condition that. is bounded.
Does compact mean bounded? ›A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set.
What is the range of compactness? ›
Fraction Kept Compactness
Scores range from 0 to 1 where 0 is least compact and 1 is most compact.
Relative compactness is another property of interest. Definition: A subset S of a topological space X is relative compact when the closure Cl(x) is compact. Note that relative compactness does not carry over to topological subspaces.
How do you measure compactness of a shape? ›For 3D shapes, the compactness C can be measured by the ratio ( area 3 ) / ( volume 2 ) , which is dimensionless and minimized by a sphere. Thus, for a sphere: A = 4 π r 2 and V = ( 4 / 3 ) π r 3 .
What is compact set with example? ›Definition 12.1. A set S⊆R is called compact if every sequence in S has a subsequence that converges to a point in S. One can easily show that closed intervals [a,b] are compact, and compact sets can be thought of as generalizations of such closed bounded intervals.
How do you show compactness? ›By scaling, any closed bounded interval is compact. Thus, a product of closed bounded intervals (i.e. a closed bounded rectangle) is compact. Any closed and bounded set is contained as a closed subset of closed and bounded intervals. Since any closed subset of a compact space is compact, this completes the proof.
What kind of word is compact? ›Adjective The drill has a compact design. the apartment's compact floor plan The cabin was compact but perfectly adequate.
What is compact method? ›The compact method in Ruby returns an array that contains non-nil elements. This means that when we call this method on an array that contains nil elements, it removes them. It only returns the other, non-nil, elements of the array.
What is the compactness of a circle? ›A circle is the most compact shape, and by definition above, it will have a compactness value of 1.