25.1 Set Theory: Problems and Paradoxes

1

25.2 TOPOLOGY

Topology, that part of geometry concerned with the properties of figures invariant under transformations that are continuous and have continuous inverses, grew from various roots in the nineteenth century to become a full-fledged division of mathematics by early in the twentieth. The two branches of the subject to be considered here are point-set topology, concerned with the properties of sets of points of some abstract “space,” and combinatorial topology, concerned with how geometrical objects are built up out of certain well-defined “building blocks.”

Point-set topology grew out of the studies of Cantor of sets of real numbers. As such, its central aim is to provide an appropriate context for generalizing such properties of the real numbers as the Bolzano-Weierstrass property and the Heine-Borel property. Both of these are closely related to the important modern notion of compactness, a notion central in the generalization of the theorem on the existence of a maximum of a continuous function on a closed interval. The Bolzano-Weierstrass property is the property that every bounded infinite set of real numbers contains at least one point of accumulation, a point in every open interval of which there is another point of the set. In other words, this property asserts the “completeness” of the real numbers. We considered Bolzano’s proof of this result in Section 22.1.3.

The Heine-Borel property, formulated by Emile Borel (1871–1956) in 1894, states that if an infinite set A of intervals covers a finite closed interval B of real numbers, in the sense that every number of B lies in the interior of at least one element of A, then there is a finite subset of A with the same property. (Although Heine had used this result implicitly in the1870s, it was Borel who first stated and proved this theorem with respect to a countable setA. Henri Lebesgue (1875–1941) generalized it to arbitrary infinite collections A in 1904.) Borel’s proof is by contradiction. Let A = {A1, A2, . . . , Am, . . .}. If the conclusion is not true, then for every n there is a point bn ∈ B such that bn is not in Ai for every i ≤ n. If onebisects the interval B , the same statement is true for at least one of the halves. If one continues this bisection process, one obtains a decreasing nested sequence of closed intervals Bi , each

B

25.2 Topology 883

BIOGRAPHY

Grace Chisholm Young (1868–1944)

orn Grace Chisholm at Haslemere, near London, she was educated at home and then entered Girton College, Cam-

bridge, the first English institution where women could receive a university education. Having attained a superior

score on the Cambridge Tripos exam in 1892, she decided to go to Go¨ ttingen to continue her studies because

there was no pos- sibility of advanced study in England. Felix Klein was willing to accept women students, but only

after he had assured himself through a personal interview that they would succeed. (There were other members of the

faculty who objected to admitting women under any conditions.) In any case, Grace Chisholm earned her PhD in 1895, being the first woman to receive a Ger- man doctorate in mathematics through the regular procedure. In 1896, she

married William Young, an English mathemati-cian who had been her tutor at Girton.

The Youngs spent the next 44 years in a partnership fruitful both in mathematics and in children (six). Although most of the more than 200 mathematical papers and books that ensued were in William Young’s name, Grace had a major role in their production. As William noted in an article of 1914, he had discussed the major idea of the work with his wife, and Grace had elaborated the argument and put the paper into publishable form. Their daughter wrote that her father could only generate ideas when he was stimulated by a sympathetic audience. Not only did Grace provide this audience, but she also had the initiative and stamina to complete the various undertakings her husband proposed. William died at their home in Switzerland after the outbreak of the Second World War left Grace in England. She died in 1944, just before she was to receive an honorary degree from the Fellows of Girton College. Two of their sons as well as one of their granddaughters alsobecame mathematicians.

of which has the same property as B itself. But ∩Bi contains a point p. By hypothesis, p isin the interior of Ak for some k, so Ak must contain one of the intervals Bi , a contradiction.

The key to Borel’s proof is what is often called the nested interval property, that the intersection of a nested family of closed intervals contains a point. It was this result that was later to be abstracted into the earliest definition of compactness. It was also this result that was among the earliest presented in the first systematic exposition of set theory as a whole, the 1906 The Theory of Sets of Points by William Young (1863–1942) and Grace Chisholm Young (1868–1944).

25.2.1 The Youngs and The Theory of Sets of PointsThe text of the Youngs deals with sets of points on the real line or in the real plane and gives explicit definitions of many fundamental concepts that were later to be generalized. For example, a point x belonging to an interval “unclosed” at both ends is said to be an internal point of the corresponding closed interval. A point L is said to be a limit point of a given set of real numbers if inside every interval containing L as an internal point there is a point of the set other than L. A set that contains all its limit points is said to be closed, while one that does not is said to be unclosed or open. (Note that this is not the definition of “open” in use today.) The Youngs then reformulated the Bolzano-Weierstrass and Heine-Borel theorems in terms of these definitions and presented proofs.

884 Chapter 25 Aspects of the Twentieth Century and Beyond

The Youngs next generalized the notion of “interval” in the line to that of “region” in the plane by considering the latter to be generated by a set of triangles contained in it and then generalized the notion of limit point by replacing “interval” with “region.” They noted further that, in analogy with the properties of an interval, a region divides the plane into three disjoint sets: internal points (those internal to at least one generating triangle), boundary points (those other than internal points that are limit points of internal points), and external points (those that are neither internal points nor boundary points). They then easily stated and proved the generalizations of the Bolzano-Weierstrass and Heine-Borel theorems to the plane.

Another fundamental notion in modern topology, the idea of connectedness, stems from some considerations of Cantor. Cantor, as part of his work in dealing with the “continuum,” the entire set of real numbers, tried to characterize this set. As part of this attempt, he found it necessary to define what it meant for the set to be in one piece. Because he saw this idea in terms of the minimal distance between points of the set being 0, he defined “connected” in terms of distance: A set T is connected if given any two points p and q in it and any positive number , there is a finite number of points t1, t2, . . . , tn in T such that the distances pt1, t1t2, . . . , tnq are all less than . The Youngs, among others, realized that it would be better to give a definition not using distance, but purely in set-theoretic terms, so they translated this notion as follows: “A set of points such that, describing a region in any manner round each point and each limiting point of the set as internal point, these regions always generate a single region, is said to be a connected set provided it contains more than one point.”7

Using this definition, the authors then proved that a set is connected if and only if it cannot be divided into two closed components without common points.

25.2.2 Fre´ chet and Function SpacesIn the same year as the Youngs’ text appeared, Maurice Fre´chet (1878–1973) began the process of generalizing results on points in the plane to more general contexts. In his disser- tation dealing with the theory of functionals, functions operating on certain sets of functions, he had to be able to decide when two functions were “close” to each other and therefore de- cided “to establish systematically certain fundamental principles of the Functional Calculus, and then to apply them to certain concrete examples.” By doing so, “one often gains thereby from seeing more clearly that which was essential in the demonstration, . . . from the sim- plification, and in the freeing [of the proofs] from that which only depends on the particular nature of the elements considered.”8 In other words, Fre´chet decided to reconsider the basic notions of the topology of the real line in terms of arbitrary sets and then apply these notions to his particular sets of interest. He realized that he could prove numerous results of topology in a general context once and for all and then apply them in concrete instances rather than prove the same result over and over again. He especially wanted to answer questions about convergence of functionals to limits and determine the circumstances under which the limit functional has the same properties of the functionals of which it is the limit.

Fre´chet thus began, not by defining “limit,” but by characterizing the notion by means of axioms. That is, a set E belongs to L, the class for which limits are defined, if given anyinfinite subset {ai } of E it is possible to determine whether or not a unique element a exists having appropriate properties. This element is to be the limit of {ai }. The limit element a

25.2 Topology

885is subject to the conditions that if each ai = a, then the limit is a, and if a is the limit of{ai }, then it is also the limit of any subsequence of {ai }. With this abstract notion of a limit,

886 Chapter 25 Aspects of the Twentieth Century and Beyond

Fre´chet stated various definitions, some of which had already been considered by Cantor: The derived set E of E is the set of its limit elements. The set E is closed if E ⊆ E and perfect if E = E. The point a is an interior point of E if a is not the limit of any sequencenot in E.

As part of his study of functionals, Fre´chet wanted to generalize the Heine-Weierstrass result on the existence of a maximum and minimum of a continuous function on a closed interval so that it could apply to function spaces. To accomplish this generalization, he took the central idea of the proof of the Heine-Borel theorem as a definition in a brief note of 1904: “We call a compact set every set E such that there always exists at least one element common to every infinite sequence of subsets E1, E2, . . . , En, . . . of E, if these (possessing at leastone element each) are closed and each is contained in the preceding one.”9 Fre´chet then provedthat a necessary and sufficient condition for a set E to be compact was that every infinite subset F had at least one limit element in E, an element that is the limit of some sequence of distinct points of F . Noting that compact sets by his definition had properties analogous to those of closed and bounded sets in space, Fre´chet was able to generalize the Weierstrass result, using for his definition of continuity the property today generally known as sequentialcontinuity: A function f is continuous at a in the closed set E if limn→∞ f (an) = f (a) for all sequences {an} in E that converge to a. Interestingly, in his dissertation of 1906, Fre´chet tookfor his definition of “compact” the necessary and sufficient condition above rather than theintersection property, adding the statement that any finite set will also be considered compact.

Having shown that the concept of a distance was not necessary to develop various familiar notions, Fre´chet proceeded to reintroduce that notion in a more general setting. Namely, he considered a subclass E of the class L consisting of members E in which can be defineda metric, a real-valued function (a, b) satisfying (1) (a, b) = (b, a) ≥ 0; (2) (a, b) = 0 if and only if a = b; and (3) (a, b) ≤ (a, c) + (c, b) for any three elements a, b, c of E. Using the metric, Fre´chet defined a Cauchy sequence to be any sequence {an} such thatfor every > 0 there is an m so that for all p > 0, we have (am, am+p) < . Limit elements and the concepts associated with them are then defined in terms of the metric and Cauchysequences. In particular, Fre´chet considered the subclass of normal sets consisting of sets that are perfect, separable (contain a countable dense subset), and in which every Cauchy sequence has a limit (complete, in modern terminology). It is for sets of this type that he could prove a generalization of the Heine-Borel theorem: A normal set E is compact if andonly if for every collection G of sets {I } such that every element of E is in the interior of atleast one member of that collection, there exists a finite subset of G with the same property.

Fre´chet gave a number of examples of spaces with metrics for which his general theorems held, including examples of sets of functions. One such example was the set of real functions continuous on a given closed interval. Here the metric was the maximum norm

(f, g) = max |f (x) − g(x)|,x∈I

under which Fre´chet proved that the set was normal. A second example was the set of all sequences of real numbers x = {x1, x2, . . . , } with the metric

∞25.2 Topology

887

(x, y) = 1 |xp − yp |

.p=1

p! 1 + |xp − yp |

Fre´chet noted that this metric is better than a more standard one, (x, y) = maxp |xp − yp

|,in the sense that the former always gives a finite distance while the latter does not. Again,Fre´chet showed that this set is normal and, furthermore, that the same is true for the set of real functions defined on it with an appropriate metric.

25.2.3 Hausdorff and Topological SpacesIt was Felix Hausdorff (1868–1942) who was able to give a full axiomatization of the notion of a topological space derived from standard properties of sets of real numbers. He described this axiomatization in his 1914 text, Grundzu¨ ge der Mengenlehre (Foundations of Set Theory). Although his axioms and definitions have been modified since 1914 and numerous subsidiary definitions and concepts have been introduced, the basics of point-set topology as currently taught are all to be found in this fundamental work.

Hausdorff noted that there are three basic concepts by means of which one can base a general theory of topological spaces: the notions of distance, neighborhoods, and limits. Fre´chet had in effect used both the first and the third. Hausdorff noted that if one begins with the concept of distance, one can derive the other two, while if one begins with that of neighborhood, one can define the notion of limits. In general, however, one cannot reverse these procedures. Although the method one chooses, according to Hausdorff, is a matter of taste, he decided to begin with the concept of neighborhood to define the notion of a topological space. Such a space, today known as a Hausdorff space, is a set E to eachelement x of which is associated a collection of subsets {Ux } of E, called neighborhoods,which satisfy the following axioms:

1. Each point x belongs to at least one neighborhood Ux , and every neighborhood Uxcontains x.

2. If Ux , Vx are two neighborhoods of the same point x, then their intersection also contains a neighborhood of that point.

3. If y ∈ Ux , then there is a neighborhood Uy such that Uy ⊆ Ux .4. For two different points x, y there are two neighborhoods Ux , Uy whose intersection is

empty.10

After showing that a metric space, with the neighborhoods defined by Ux = {y|(y, x) < ρ} for every real number ρ, satisfies the axioms given, Hausdorff developed the same basic theory as had Fre´chet. The major change was that the central concept was now the domain or, in more modern terms, the open set. For Hausdorff, a domain A is a subset of E containing only interior points, where the latter are defined as points x that have a neighborhood Ux contained in A. That the entire set E and each of the neighborhoods Ux

are domains follows from axioms 1 and 3. Hausdorff also showed that the union of arbitrarily many domains and the intersection of finitely many are domains. A closed set, on the other hand, is one that contains all of its accumulation points. Hausdorff then showed that the closed sets areprecisely the complements of the domains.

Using Fre´chet’s limit definition for compact sets, Hausdorff proved the intersection prop- erty of nested, closed, compact sets and then a generalization of the Heine-Borel theorem, modifying the original proof only slightly. He also gave a new definition of limit point and convergence general enough to apply in any topological space: The point x is a

limit of the infinite set A, if every neighborhood Ux of x contains all but a finite number of points of A.

Further, because any set A either has a single limit x or none at all, one writes in the first case that x = lim A or that A converges to x.11 Among the other new definitions was one forconnectedness: “A set A differing from the empty set is connected if it cannot be divided into two disjoint, non-null, sets both closed relative to A [that is, such that each set is the intersec- tion with A of a closed set in the ambient space E].”12 Hausdorff noted that because closed sets are the complements of domains, he could equally well have specified in the definition that the components needed to be domains.

Because the idea of a neighborhood was the point of departure for Hausdorff’s develop- ment, it is not surprising that he defined continuity using that concept. Namely, he took thestandard − δ definition of continuity for functions of a real variable, noted that this def-inition used neighborhoods on the real line, and then translated it into a general definition for topological spaces: “The function y = f (x) is called continuous at the point a, if for each neighborhood Vb of the point b = f (a) there exists a neighborhood Ua of the point a, whose image lies in Vb; i.e., f (Ua ) ⊆ Vb.”13 Hausdorff then generated an equivalent defini- tion: The function f : A → B is continuous at a if and only if, for each subset Q of B for which b = f (a) is an interior point, the inverse image f −1(Q) also contains the point a as aninterior point. Using this definition, it is easy to prove that a continuous function preservesconnectedness and compactness. Hausdorff naturally noted that the first of these facts im- plies the intermediate value theorem while the second implies the existence of a maximum and minimum value for a continuous function on a closed interval. In other words, Hausdorff showed that a topological space is the natural setting for these classical results about functions of a real variable.

25.2.4 Combinatorial TopologyCombinatorial topology grew out of the study of the idea of multiple connectivity of a surface in space, an idea developed by Riemann in his work on the integration of differentials. This idea was further refined by the physicists of the mid-nineteenth century because it turned out to be important in fields such as fluid dynamics and electromagnetism. It was Enrico Betti (1823–1892), however, who in 1871 generalized the idea of multiple connectedness to n- dimensional spaces by using hypersurfaces without boundaries as the analogues of Riemann’s closed curves. Betti furthermore applied this idea to study the integration of differential forms over spaces of dimension n. The differences in connectivity of two surfaces were a way of telling that the surfaces were essentially “different,” that there could be no continuous invertible function from one to the other. To deal with this method of distinguishing surfaces, however, Poincare´ developed the idea of homology.

In his fundamental paper, Analysis Situs, of 1895 and four years later in a supplemen- tary work, Poincare´ made the following definition: A homology relation exists among p- dimensional subvarieties ν1, ν2, . . . , νr of an n-dimensional variety V , written

ν1 + ν2 + . . . + νr ∼

0,

if for some integer k, the set consisting of k copies of each of the νi constitutes the complete boundary of a (p + 1)-dimensional subvariety W . (A “variety” for Poincare´—now generallycalled a manifold—was the generalization to higher dimensions of a curve in one dimension or a surface in two dimensions and was usually thought of as being defined, at least locally, either as the set of zeros of an appropriate system of functions or

parametrically as the image of

a certain set of such functions.) Poincare´ introduced “negatives” of varieties by considering orientation. That is, −ν is the same variety as ν but with the opposite orientation. As anexample of a homology relation, let ν1 and ν2 be the outer and inner boundaries, respectively, of the ring in Figure 25.3 with the indicated orientations. Then ν1 and ν2 together formthe complete boundary of the ring, and, since −ν2 has the opposite orientation as ν2, the relation ν1 − ν2 ∼ 0 holds. Poincare´ further observed that the varieties in homology relationscan be added, subtracted, and multiplied by integers and therefore was able to call a set ofvarieties linearly independent if there were no homology relation among them with integer coefficients.

FIGURE 25.3

Homology relations

ν2

ν1

To clarify the notion of multiple connectedness, Poincare´

went on to define the p-

dimensional Betti number Bp of a variety V to be one more than the maximum number of linearly independent, closed, p-dimensional subvarieties, where a closed variety is one without a boundary. Thus, according to Poincare´ the 1-dimensional Betti number of the ring in Figure 25.3 is 2, while that of the double ring is 3. On the other hand, the 1-dimensional Betti number of the disc is 1. Poincare´ applied his notion of homology to the study of integra- tion over varieties of various dimensions and also attempted to prove a duality theorem, thatfor compact, connected, orientable manifolds of dimension n, the relationship Bp = Bn−pheld for 1 ≤ p ≤ n − 1. Poincare´’s proof, however, and even his definition, have flaws in theeyes of modern readers. It took about 20 years to construct a homology theory containingPoincare´’s basic ideas, which was rigorous by today’s standards.

The modern theory of homology was developed by several mathematicians in the early twentieth century. One major simplification of Poincare´’s ideas was the consideration of p- dimensional submanifolds not as solutions to systems of equations but as being formed from certain simple p-dimensional manifolds, each of which was the continuous image of

a p- dimensional “triangle.” The appropriate definitions were completely worked out by James W. Alexander (1888–1971) by 1926 when he defined a p-simplex to be the p-dimensional analogue of a triangle and a complex to be a finite set of simplexes such that no two had an interior point in common and such that every face of a simplex of the set was also a simplex

P

BIOGRAPHY

Henri Poincare´ (1854–1912)

oincare´, like Hilbert, was a universal mathematician, one who contributed to virtually every area of mathematics,

including physics and theoretical astronomy. He was born into an upper-middle-class family, many members of whom

per- formed various services to the French government. Poincare´ displayed a strong interest in mathematics from an

early age and won a first prize in the mathematics competition for stu- dents in all of the French lyce´es. In

1873, he entered theE´ cole Polytechnique and, after receiving his doctorate in 1879,

entered a university career, first at the University of Caen and then, in 1881, at the University of Paris. Toward the end of his life, he turned to popularization and wrote several books emphasizing the importance of science and mathematics, in- cluding such works as Science and Hypothesis, The Value of Science, and Science and Method. It was in the latter work that Poincare´ described the psychology of discovery in mathemat- ics, stressing the subconscious as a central factor in mathemat-ical creativity (Fig. 25.4).

FIGURE 25.4

Poincare´ on a French stamp

of the set. An elementary i -chain of a complex was defined to be an expression of the form±V0V1 . . . Vi , where the V ’s are vertices of an i -simplex. The expression changes sign

uponany transposition of the V ’s, thus giving each chain an orientation. An elementary i -chainwas then an i -dimensional “face” of a p-simplex, while an arbitrary i -chain was a linear combination of elementary i -chains with integer coefficients. As an example, the tetrahedron with vertices V0, V1, V2, V3 is a 3-simplex, while it together with its four faces (each a 2- simplex), its four edges (each a 1-simplex), and its four vertices (each a 0-simplex), form a complex. The face V0V1V2 is then an elementary 2-chain of the 3-simplex. Alexandernext defined the boundary of the elementary i -chain K = V0Vi . . . Vi to be the (i − 1)-chain K =

(−1)s V0 . . . Vˆ

s . . . Vi and extended this to arbitrary i -chains by linearity. Thus,the boundary of V0V1V2 is V1V2 − V0V2 + V0V1 (Fig. 25.5). An easy calculation with this

FIGURE 25.5 V3

The boundary of a tetrahedron

V0 V2

V1

896 Chapter 25 Aspects of the Twentieth Century and Beyond

example shows that the boundary of the boundary is zero, and one can show that this result is true in general.

Alexander applied his definition of homology to closed chains (cycles), chains whose boundary is zero. Namely, a closed chain K is homologous to zero, K ∼ 0, if it is the boundary of a chain L. Two chains K and K ∗ are homologous, K ∼ K ∗, if K − K ∗ is homologous tozero. The pth Betti number of a complex is then the maximum number of closed p-chains that are linearly independent with respect to boundary, that is, such that no linear combination is homologous to zero. (Note that this number is one less than the number according to Poincare´’s original definition.)

With a commutative operation (“addition”) having an inverse being considered on the set of closed chains, it should be clear to modern readers that there is a group hiding among Alexander’s definitions. Mathematicians of the 1920s saw this as well. But before discussing the applications of group theory to topology, we need first to consider the growth of algebra in general in the early part of the twentieth century.

de.