See homology and cohomology[III.38]
III.9 Compactness and Compactification
Terence TaoIn mathematics, it is well-known that the behavior of finite sets and the behavior of infinite sets can be rather different. For instance, each of the following statements is easily seen to be true wheneverXis a finite set but false wheneverXis an infinite set.
All functions are bounded. If f : X → Ris a real- valued function onX, thenf must be bounded (i.e., there exists a finite numberMsuch that|f (x)|M for allx∈X).
All functions attain a maximum.Iff:X→Ris a real- valued function onX, then there must exist at least one point x0 ∈ X such thatf (x0) f (x) for all x∈X.
All sequences have constant subsequences.Ifx1, x2, x3,· · · ∈Xis a sequence of points inX, then there must exist a subsequence xn1, xn2, xn3, . . . that is constant. In other words,xn1 =xn2 = · · · = cfor somec ∈X. (This fact is sometimes known as the infinite pigeonhole principle.)
The first statement—that all functions on a finite set are bounded—can be viewed as a very simple exam- ple of alocal-to-global principle. The hypothesis is an assertion of “local” boundedness: it asserts that|f (x)|
is bounded for each pointx∈X separately, but with a bound that may depend onx. The conclusion is that of “global” boundedness: that|f (x)|is bounded by a singleboundMfor allx∈X.
So far we have viewed the object X only as a set.
However, in many areas of mathematics we like to endow our objects with additional structures, such as a topology[III.90], ametric[III.56], or agroup struc- ture[I.3 §2.1]. When we do this, it turns out that some
objects exhibit properties similar to those of finite sets (in particular, they enjoy local-to-global principles), even though as sets they are infinite. In the categories of topological spaces and metric spaces, these “almost- finite” objects are known ascompact spaces. (Other cat- egories have “almost-finite” objects as well. For exam- ple, in the category of groups there is a notion of a pro-finite group; forlinear operators[III.50] between normed spaces[III.62] the analogous notion is that of acompact operator, which is “almost of finite rank”;
and so forth.)
A good example of a compact set is the closed unit intervalX=[0,1]. This is an infinite set, so the previ- ous three assertions are all false as stated forX. But if we modify them by inserting topological concepts such as continuity and convergence, then we can restore these assertions for[0,1]as follows.
All continuous functions are bounded. Iff:X→Ris a real-valued continuous function onX, thenfmust be bounded. (This is again a type of local-to-global principle: if a function does not vary too much locally, then it does not vary too much globally.)
All continuous functions attain a maximum. Iff : X → Ris a real-valued continuous function onX, then there must exist at least one pointx0∈Xsuch thatf (x0)f (x)for allx∈X.
All sequences have convergent subsequences. If x1, x2,x3,· · · ∈Xis a sequence of points inX, then there must exist a subsequencexn1, xn2, xn3, . . .that converges to some limitc ∈ X. (This statement is known as theBolzano–Weierstrass theorem.) To these assertions we can add a fourth (which, like the others, has a rather trivial analogue for finite sets).
All open covers have finite subcovers.IfV is a col- lection of open sets and the union of all these open sets containsX (in which caseV is called anopen cover of X), then there must exist a finite subcol- lectionVn1, Vn2, . . . , Vnk of sets inV that still covers X.
All four of these topological statements are false for sets such as the open unit interval(0,1)or the real lineR, as one can easily check by constructing simple counterexamples. TheHeine–Borel theoremasserts that whenXis a subset of a Euclidean spaceRn, the above statements are all true whenX is topologically closed and bounded, and all false otherwise.
The above four assertions are closely related to each other. For instance, if you know that all sequences in X contain convergent subsequences, then you can quickly deduce that all continuous functions have a maximum. This is done by first constructing amaximiz- ing sequence—a sequence of pointsxninX such that f (xn)approaches the maximal value off(or, more pre- cisely, its supremum)—and then investigating a conver- gent subsequence of that sequence. In fact, given some fairly mild assumptions on the space X (e.g., that X is a metric space), one can deduce any of these four statements from any of the others.
To oversimplify a little, we say that a topological spaceXiscompact if one (and hence all) of the above four assertions holds forX. Because the four assertions are not quite equivalent in general, the formal defini- tion of compactness uses only the fourth version: that every open cover has a finite subcover. There are other notions of compactness, such as sequential compact- ness, for example, which is based on the third version, but the distinctions between these notions are technical and we shall gloss over them here.
Compactness is a powerful property of spaces, and it is used in many ways in many different areas of math- ematics. One is via appeal to local-to-global principles:
one establishes local control on a function, or on some other quantity, and then uses compactness to boost the local control to global control. Another is to locate maxima or minima of a function, which is particularly useful in thecalculus of variations[III.94]. A third is to partially recover the notion of a limit when deal- ing with nonconvergent sequences, by accepting the need to pass to a subsequence of the original sequence.
(However, different subsequences may converge to dif- ferent limits; compactness guarantees the existence of a limit point, but not its uniqueness.) Compactness of one object also tends to beget compactness of other objects; for instance, the image of a compact set under a continuous map is still compact, and the product of finitely many or even infinitely many compact sets continues to be compact. This last result is known as Tychonoff’s theorem.
Of course, many spaces of interest are not compact.
An obvious example is the real lineR, which is not com- pact, because it contains sequences such as 1,2,3, . . . that are “trying to escape” the real line and that do not leave behind any convergent subsequences. However, one can often recover compactness by adding a few more points to the space: this process is known ascom- pactification. For instance, one can compactify the real
line by adding one point at each end: we call the added points+∞and−∞. The resulting object, known as the extended real line[−∞,+∞], can be given a topology in a natural way, which basically defines what it means to converge to+∞ or to −∞. The extended real line is compact: any sequencexnof extended real numbers will have a subsequence that either converges to+∞, or converges to−∞, or converges to a finite number. Thus, by using this compactification of the real line, we can generalize the notion of a limit to one that no longer has to be a real number. While there are some draw- backs to dealing with extended reals instead of ordi- nary reals (for instance, one can always add two real numbers together, but the sum of+∞and−∞is unde- fined), the ability to take limits of what would other- wise be divergent sequences can be very useful, par- ticularly in the theory of infinite series and improper integrals.
It turns out that a single noncompact space can have many different compactifications. For instance, by the device ofstereographic projection, one can topologi- cally identify the real line with a circle that has a sin- gle point removed. (For example, if one maps the real numberxto the point(x/(1+x2), x2/(1+x2)), then Rmaps to the circle of radius12and center(0,12), with the north pole (0,1)removed.) If we then insert the missing point, we obtain theone-point compactification R∪ {∞}of the real line. More generally, any reason- able topological space (e.g., a locally compact Haus- dorff space) has a number of compactifications, rang- ing from the one-point compactificationX∪{∞}, which is the “minimal” compactification as it adds only one point, to theStone– ˇCech compactificationβX, which is the “maximal” compactification and adds an enormous number of points. The Stone–ˇCech compactificationβN of the natural numbers Nis the space ofultrafilters, which are very useful tools in the more infinitary parts of mathematics.
One can use compactifications to distinguish be- tween different types of divergence in a space. For instance, the extended real line[−∞,+∞]distinguishes between divergence to+∞and divergence to−∞. In a similar spirit, by using compactifications of the plane R2such as theprojective plane[I.3 §6.7], one can dis- tinguish a sequence that diverges along (or near) the x-axis from a sequence that diverges along (or near) the y-axis. Such compactifications arise naturally in situa- tions in which sequences that diverge in different ways exhibit markedly different behavior.
Another use of compactifications is to allow one to view one type of mathematical object rigorously as a limit of others. For instance, one can view a straight line in the plane as the limit of increasingly large circles by describing a suitable compactification of the space of circles that includes lines. This perspective allows us to deduce certain theorems about lines from analo- gous theorems about circles, and conversely to deduce certain theorems about very large circles from theo- rems about lines. In a rather different area of mathe- matics, the Dirac delta function is not, strictly speaking, a function, but it exists in certain (local) compactifica- tions of spaces of functions, such as spaces of mea- sures[III.55] ordistributions[III.18]. Thus, one can view the Dirac delta function as a limit of classical func- tions, and this can be very useful for manipulating it.
One can also use compactifications to view the continu- ous as the limit of the discrete: for instance, it is possi- ble to compactify the sequenceZ/2Z,Z/3Z,Z/4Z, . . .of cyclic groups in such a way that their limit is the circle groupT=R/Z. These simple examples can be general- ized to much more sophisticated examples of compact- ifications, which have many applications in geometry, analysis, and algebra.
III.10 Computational Complexity Classes
One of the basic challenges of theoretical computer sci- ence is to determine what computational resources are necessary in order to perform a given computational task. The most basic resource istime, or equivalently (given the hardware) the number of steps needed to implement the most efficient algorithm that will carry out the task. Especially important is how this time scales up with the size of the input for the task: for instance, how much longer does it take to factorize an integer with 2ndigits than an integer with n digits?
Another resource connected with the feasibility of a computation ismemory: one can ask how much stor- age space a computer will need in order to implement an algorithm, and how this can be minimized. Acom- plexity classis a set of computational problems that can be performed with certain restrictions on the resources allowed. For instance, the complexity classP consists of all problems that can be performed in “polynomial time”: that is, there is some positive integerksuch that if the size of the problem isn(in the example above, the size was the number of digits of the integer to be fac- torized), then the computation can be carried out in at
mostnksteps. A problem belongs toPif and only if the time taken to solve it scales up by at most a constant factor when the size of the input scales up by a con- stant factor. A good example of such a problem is mul- tiplication of twon-digit numbers: if you use ordinary long multiplication, then replacingnby 2n increases the time taken by a factor of 4.
Suppose that you are presented with a positive inte- gerxand told that it is a product of two primespand q. How difficult is it to determinep and q? Nobody knows, but one thing is easy to see: if you are toldp andq, then it is not hard (for a computer, at any rate) to check that pq really does equal x. Indeed, as we have just seen, long multiplication takes polynomial time, and comparing the answer with x is even eas- ier. The complexity classNPconsists of those compu- tational tasks for which a correct answer can bever- ified in polynomial time, even if it cannot necessar- ily befoundin polynomial time. Remarkably, although this is a fundamental distinction, nobody knows how to prove thatP = NP: this problem is widely consid- ered to be the most important in theoretical computer science.
We briefly mention two other important complexity classes.PSPACEconsists of all problems that can be solved using an amount of memory that grows at most polynomially with the size of the input. It turns out to be the natural class associated with reasonable compu- tational strategies for games such as chess. The com- plexity classNCis the set of all Boolean functions that can be computed by a “circuit of polynomial size and depth at most a polynomial in logn.” This last class is a model for the class of problems that can be solved very rapidly using parallel processing. In general, com- plexity classes are often surprisingly good at character- izing large families of problems with interesting and intuitively recognizable features in common. Another remarkable fact is that almost all complexity classes have “hardest problems” within them: that is, problems for which a solution can be converted into a solution for any other problem in the class. These problems are said to becompletefor the class in question.
These issues, as well as several other complexity classes, are discussed incomputational complexity [IV.20]. A vast number of further classes can be found at
http://qwiki.stanford.edu/wiki/Complexity_Zoo along with a brief definition of each.