Connected Setsand the AMS, 1901–1921David E. Zitarelli

Chapter 1 of Kelley’s famous bookGeneral Topology introduces the mostfundamental concepts of a topologicalspace. One such notion is defined asfollows [1]:

A topological space (X, τ) is con-nected if and ony if X is not theunion of two nonvoid separatedsubsets, where A and B are sepa-rated in X if and only if A∩ B = �and A∩ B = �.

As usual, Y denotes the closure of a subset Yof X.

Kelley’s book has been a staple for severalgenerations of graduate students, many of whommust have wondered what this formal definitionhad to do with their intuitive notion of a connectedset. Frequently, such queries can be answered byan historical investigation, and the aim here isto trace the development of the formal conceptof a connected set from its origins in 1901 untilits ultimate ascension into the ranks of mathe-matical concepts worthy of study for their ownsake twenty years later. Much of this developmenttook place at the University of Chicago underE. H. Moore, and the evolution of connected setsexemplifies one specific way in which ideas thatgerminated there would be promulgated by hisdescendants. Although Moore exerted little directinfluence, his department’s core of outstandinggraduate students, like the well-known OswaldVeblen, and its cadre of small-college instructorsand high school teachers from across the countrywho pursued degrees during summer sessions,

David E. Zitarelli is associate professor of mathematics atTemple University. His z-mail address is zit@temple.edu.

like the lesser-known N. J. Lennes, would play adecisive role.

An overarching theme is how the rapidly evolv-ing AMS abetted this development in two ways.For one, local and national meetings providedvenues where researchers could present their workand keep abreast of the progress of others. Foranother, the two AMS publications, the Bulletin andthe Transactions, provided outlets for publishingthese findings. Neither journal had widespreadreadership across the Atlantic so, as we will see, thestudy of connected sets would proceed in Europeindependently of advances in America. Initially,European initiatives appeared in a Polish journaland were based on a classic book by Hausdorff,but ultimately the two schools of topology wouldinteract symbiotically over connected sets. Oneof the contributors who engendered the ensuinginternational collaboration was Anna Mullikin, asecond-generation Moore descendant who becamethe first American to publish a paper devoted toconnected sets. We end our account by examiningone of her chief examples, Mullikin’s nautilus, toillustrate the power of the general definition firstproposed by N. J. Lennes. We begin by introducingthe latter’s life and work, emphasizing the formu-lation of connected sets he announced fifty yearsbefore the appearance of Kelley’s classic.

The Pioneer N. J. LennesIt seems appropriate that the person who pioneeredthe modern definition of a connected set wouldbe a pioneer himself in a geographic sense. NelsJohann Lennes (1874–1951) came to the U.S. fromhis native Norway at age sixteen and, like manyScandinavians, settled in Chicago. Recall that theUniversity of Chicago opened its doors in 1892;

450 Notices of the AMS Volume 56, Number 4

N. J. Lennes

Lennes enrolled four years later and earned a bach-elor’s degree in just two years. Upon graduation hebecame a high school teacher (1898–1907), takinggraduate courses at Chicago all the while. With hisPh.D. in hand he then taught at the MassachusettsInstitute of Technology and Columbia Universityfor three years each before his pioneering move todistant Missoula, Montana, where he was professorof mathematics and head of the department from1913 until retirement in 1944. ([2] provides anoverview of his life and work; he is describedas “one of the precursors of modern abstractmathematics”.)

This biographical vignette indicates that whileLennes was a student at Chicago he rubbed elbowswith two other students who would ultimatelymake important contributions to this development,Oswald Veblen (1880–1960) and R. L. Moore (1882–1974). The mathematics department in Eckhart Hallwas a cauldron of activity, and this trio emergedduring a ten-year period that produced such leg-endary figures as L. E. Dickson (1874–1954), G. A.Bliss (1876–1951), and G. D. Birkhoff (1884–1944).

Veblen, recently profiled in the Notices [3], en-rolled at Chicago in 1900 after having spent a yearat Harvard obtaining a second bachelor’s degree. Itis known that he first impressed E. H. Moore duringthe fall 1901 seminar “Foundations of Geometry”,and that his subsequent proof of the Jordan CurveTheorem (JCT) was inspired by discussions in the

seminar. It is not so widely known that Lennesplayed a decisive role with the JCT too. In fact,Veblen’s first paper on topology reported thatLennes, in his 1903 master’s thesis, proved thespecial case of the JCT for a simple polygon [4, p.83]. Because the idea of a curve dividing a planeinto separated subsets suggests the definition of adisconnected set, as implicitly defined by Lennes,we date our history from 1901.

The Moore seminar was pivotal for the careers ofVeblen and Lennes, and as usual the AMS played amajor role. The Chicago cauldron’s mix of topologi-cal ingredients simmered at the April 1903 meetingof the Chicago Section, where Moore’s advancedgraduate students delivered noteworthy presenta-tions of research they had begun in the seminar.The twenty-two-year-old Veblen announced majorresults that would appear in his dissertation, one ofwhich included a proof independent of Lennes that“The boundary of a simple polygon lying entirelyin a plane α decomposes α into two regions” [5,p. 365]. Veblen was in the process of establishingthe first rigorous proof of the JCT, which he labeled“The fundamental theorem of Analysis Situs” in apaper read at the AMS meeting that preceded theInternational Congress at the St. Louis World’s Fairin 1904 [4, p. 83]. The JCT remains a benchmarkof mathematical rigor today. (See [7] for a modern,computer-oriented account of this phenomenon.)

At the April 1903 meeting, Lennes, six yearsVeblen’s senior, presented results from his mas-ter’s thesis, “Theorems on the polygon and thepolyhedron”. In what was to become customary forhim, he would not submit this work for publicationfor another seven years even though he would thenassert that only “minor changes and additions havebeen made since that time” [8, p. 37]. The chiefresult was that “the polygon and polyhedron sepa-rate the plane and the three-space respectively intotwo mutually exclusive sets” [Ibid.], an affirmationof the inspiration for his definition of a connectedset in terms of separated domains.

Even though primarily engaged with teachinghigh school, Lennes continued his activity with theAMS by presenting two papers at the December1904 meeting of the Chicago Section. One dealtwith Hilbert’s theory of area and resulted in aTransactions publication the next year, reflectingan ongoing interest in geometrical topics. Theother was concerned with improper definite inte-grals, resulting in a paper in the American Journal.During this time he also published a paper on uni-form continuity in the Annals, his first publicationnot read first before an AMS audience. Moreover,Lennes submitted a paper on real function theorythat was read by title at the annual AMS summermeeting in September 1905.

The focus of Nels Lennes’s research programwas analysis, a topic he was pursuing in earnestwith Oswald Veblen, who remained at Chicago for

April 2009 Notices of the AMS 451

two years after receiving his Ph.D. in 1903. It isknown that Veblen played a central role in men-toring R. L. Moore and directing his dissertation,but the symbiotic relationship between Veblen and

[Lennes’sdefinition]

represented adramatic shift

from thegeometric,

constructiveapproach…

Lennes seems to have escapedattention. Their joint activitycan be seen in Veblen’s disser-tation, where he expressed his“deep gratitude to Professor E. H.Moore…and also to Messrs. N. J.Lennes and R. L. Moore, whohave critically read parts of themanuscript” [5, p. 344].

Veblen and Lennes alsocollaborated on a textbookaimed to “be used as a ba-sis for a rather short theoreticalcourse on real functions” [9,p. iii]. Three years before thetext appeared in 1907, Veblenrevealed that “The equivalence[of the Heine-Borel theorem with

the Dedekind cut proposition] in question suggest-ed itself to Mr. N. J. Lennes and myself while wewere working over some elementary propositionsin real function theory” [6, p. 436]. The centralfeature of the book was the “Heine-Borel proper-ty”, today called compactness, which Lennes haddiscussed at an AMS meeting in December 1905.Moreover, the final chapter, described as “moreadvanced in character than the other chaptersand intended as an introduction to the study of aspecial subject” [9, p. iii], elaborated upon Lennes’spaper on improper integrals.

Before embarking on Lennes’s formulation ofconnected sets, it is instructive to examine theintuitive approach that preceded his more gen-eral advance. This can be seen in the doctoraldissertation of L. D. Ames that was published inthe October 1905 issue of the American Journal.One of the “preliminary fundamental conceptions”he presented was the following definition, whosewording shows that set theory was still in itsformative stage [10, p. 366]:

An assemblage is connected … ifP0(x0, y0, z0) and P1(x1, y1, z1) areany two points of the assemblage,then it is possible to draw a simplecurve

x = λ(t), y = µ(t), z = υ(t),t0 ≤ t ≤ t1

having P0 and P1 as end points andsuch that all points of the curveare points of the assemblage.

Such a property is called arcwise (or pathwise)connected today.

Lewis Darwin Ames (b. 1869) was no slouch.While teaching at a normal school from 1890 to

1900, he spent the summers of 1897 and 1898taking courses at the University of Chicago. Heearned bachelor’s degrees from Missouri in 1899and Harvard in 1901. Ames remained at Harvardfor another two years, the first as an instructor andsecond as a graduate scholar, before returning tothe University of Missouri in the fall of 1903. Duringthat academic year he completed his dissertationunder the well-known analyst William Fogg Osgood(1864–1943), thus becoming the first of Osgood’sfour doctoral students and showing that researchusing the notion of a connected set was takingplace outside Chicago. It is conceivable that Lennesand Ames crossed paths during their summerstudies but no evidence supports such a link.

Just two months after Ames’s paper appeared,Lennes delivered three lectures at a Chicago Sectionmeeting of the AMS that exhibit his depth andversatility as well as the breadth of offerings withinMoore’s department. One was the work cited aboveon the Heine-Borel Theorem. Another elaborated afundamental theorem in the calculus of variations.From the present vantage point, however, the thirdwas the most important. We turn to it now.

The Genesis of Connected SetsThe remaining paper that Nels Lennes deliveredat that December 1905 meeting, “Curves in non-metrical analysis situs”, announced the earliestformulation of a connected set [11, pp. 284–5]:

A set of points is connected if inevery pair of complementary sub-sets at least one subset contains alimit point of points in the otherset.

The abstract, including the specific wording, waspublished in the March 1906 issue of the Bulletin.This marked the first time such a formulationappeared in print, but initially it elicited littleinterest. Though expressed only for subsets of aEuclidean space, the definition extends unchangedto more general spaces. Importantly, it representeda dramatic shift from the geometric, construc-tive approach championed in Ames’s paper to anabstract, nonconstructive formulation requiringproof by contradiction.

Over the next year Lennes developed his paperinto a doctoral dissertation with the expansivetitle “Curves in non-metrical analysis situs with anapplication in the calculus of variations”. It waswritten under the direction of E. H. Moore andresulted in Lennes’s Ph.D. in 1907. Yet once againLennes did not rush into print, publishing his thesisonly in 1911. He emphasized, however, that nocritical advances had taken place in the meantime,writing, “Changes made since then are entirelyunimportant” [12, p. 287]. Therefore, our analysiswill center on this paper. Here one can also see thedistinction between the older, geometric definition

452 Notices of the AMS Volume 56, Number 4

of connectedness and the abstract formulation.Initially Lennes supplied an arcwise connecteddefinition only slightly more general than the oneadopted by Ames [12, p. 293]:

An entirely open set of points issaid to be connected if for any twopoints of the set there is a brokenline connecting them which liesentirely in the set.

But then he presented the standard definitionequivalent to the one in Kelley [12, p. 303]:

A set of points is a “connectedset” if at least one of any twocomplementary subsets contains alimit-point of points in the otherset.

It is germane to point out that the paper itselfwas not devoted to connected sets per se, but tosimple arcs (curves) in nonmetric spaces. The for-mal definition of a connected set was given alongwith several other terms about midway throughthe paper. He then supplied the following criticaldefinition [12, p. 308]:

A continuous simple arc connect-ing two points A and B, A 6= B, is abounded, closed, connected set ofpoints [A] containing A and B suchthat no connected proper subset of[A] contains A and B.

In motivating this definition Lennes served no-tice of his complete understanding of the role hewas about to play, asserting, “This definition seemsto be very near the obvious intuitional meaning ofthe term ‘arc’ or ‘curve’ ” [12, p. 289]. Althoughnot referring to connected sets, his comment onarcs could apply equally well to his formulationof this important concept. Next Lennes moved tohis primary goal of proving properties of arcs inthe plane, such as the following: for every interiorpoint C of an arc AB, the arcs AC and BC are closed,are connected, and have only C in common.

As the expanded title of the paper indicates,the origin of connected sets lay in the calculus ofvariations, a subject that was one of the specialtiesat the University of Chicago, notably with OskarBolza and Gilbert Bliss. Concerning the concludingsection, Lennes wrote that “the general theory ofthe paper is applied to the problem of proving theexistence of minimizing curves in an importantclass of problems in the calculus of variations” [12,p. 290]. Moreover, he supplied a very useful historyof the need for a proper definition of the conceptof connectedness going back to G. Cantor and W. H.Young, concluding with a Veblen paper from theirChicago days together.

Lennes had an impressive résumé by the timeof his doctorate, having published six papers inevery American outlet available to him (the Annals,Transactions, Bulletin (2), American Journal, and

Monthly), as well as the book with Veblen. But itwas his definition of connected sets over the longhaul, and his definition of a continuous simplearc for a shorter period, that represent his biggestcontributions to mathematics. (As a curious aside,Kelley’s résumé included papers in the Ameri-can Journal, Duke Journal, and Proceedings of theNational Academy of Sciences when he sought em-ployment after receiving his 1940 doctorate atVirginia under G. T. Whyburn.)

After receiving his Ph.D., Lennes left his highschool post for an instructorship at MIT, wherehe remained from 1907–10 but did not publishone work. At that time MIT was fifteen years awayfrom becoming the top-notch research institutionit is today, beginning with the hiring of NorbertWiener and Dirk Struik. But then Lennes movedto Columbia (1910–13), where he experienced aspurt of nine publications, including the forty-pagepaper based on his dissertation.

In February 1911 Lennes delivered three lecturesat an AMS meeting in New York, each of whichresulted in a paper published before the year wasout. Two months later he returned to Chicago foran historic AMS meeting. Founded in 1888, theSociety remained mainly a local organization in

…the origin ofconnected sets

lay in thecalculus of

variations…

New York City until the enter-prising faculty at the upstartUniversity of Chicago inspirednational expansion six years lat-er and established a westernoutpost called the Chicago Sec-tion in 1897. Yet it was April1911 when, at the invitation ofthe Chicago Section, the Soci-ety became truly national byholding its first meeting outsideNew York City (with the excep-tion of summer affairs). AMS secretary F. N. Colegushed, “This was in many ways a remarkableoccasion…arranged [so] that this reunion of theeastern and western members should be especial-ly marked by the delivery of President Bôcher’sretiring address.… As was under these circum-stances to be expected, the meeting was in everyway a most successful and inspiring one” [13, p.505]. The meeting was successful and inspiringtoo for Lennes, who gave two lectures. The first,“Curves and surfaces in analysis situs”, showedthat his definition of a simple continuous arc(shortened here, as then, to arc) “applies with-out any change whatever to arcs in space” [13,p. 525]. The primary aim of this work, however, wasto provide a rigorous proof that a closed contin-uous surface separates space into two connectedsets.

That October, six months later, Lennes deliveredyet another paper at an AMS meeting back inNew York that extended results from the Aprilmeeting [14, p. 165]. This turned out to be the

April 2009 Notices of the AMS 453

last time he referred to connected sets in print,but it did not signal the end of his research, ashe presented four more papers at AMS meetings,two in April 1912 and two in February 1913. It istelling from this activity, however, that his careerwould take a dramatic turn when he left NewYork that fall, as only one of those presentationsresulted in a published paper. The other three wereperhaps consigned to piles in his office that would

Like Lennes, thefour

Pennsylvaniansbenefited

enormouslyfrom AMS

meetings andjournals.

remain fixed points during thethirty-one years he headed thedepartment, because once Lennesleft Columbia for the Universityof Montana later in 1913, his re-search output virtually ground toa halt, numbering only one minorpaper (in the Monthly) on the fun-damental theorem of calculus andanother on nonmathematical log-ic (in the Mathematics Teacher).Nonetheless, he resumed writ-ing textbooks at all levels, fromhigh school through college, anactivity he had begun in collabora-tion with H. E. Slaught while stillat Chicago. Ultimately Lennes’snumerous textbooks became sosuccessful commercially that the house he built oncampus provides the residence for the presidentof the University of Montana to this day. Themathematics department at the university has heldthe Lennes Exam for undergraduate students sinceshortly after his death.

Lennes’s record illustrates the vital role the AMSplayed in three distinct ways. First, the periodswhen he published papers occurred while he was inChicago and New York, the two foci of the Society.Columbia had served as AMS headquarters sinceits founding in 1888, while Chicago inspired thetransformation from a local to a national organiza-tion and became the first official section. Secondly,Lennes normally vetted his results before AMSaudiences prior to submitting them for publication.Finally, his papers often appeared in the two AMSjournals. But the time was ripe for his generaldefinition to take hold, and although he played norole in its advance, various mathematicians madeuse of it over the next ten years, most of themallied with the University of Chicago. We turn tothis development next.

The PennsylvaniansShortly after Nels Lennes went into research hi-bernation in Montana, another product of E. H.Moore awoke from a prolonged slumber in anentirely different part of the country to feast onthe savory pickings that Lennes had left behind.R. L. Moore had been a graduate student at Chicagofrom 1903–05, and Veblen’s expressed gratitude

to Lennes and Moore for critically reading Veblen’sdissertation suggests that this trio of companionscollaborated quite closely. However, after receivinghis Ph.D. in 1905, R. L. Moore endured six years ofacademic thaw, publishing only two papers, bothbased on work done in graduate school. WhereasLennes did little research after 1913, R. L. Mooretilled fertile soil at the University of Pennsylvania,resulting in seventeen papers during 1911–20,

several of which dealt with con-nected sets. As well, all three Ph.D.students he mentored at Pennmade use of Lennes’s pioneeringwork. Like Lennes, the four Penn-sylvanians benefited enormouslyfrom AMS meetings and journals.

R. L. Moore’s connection toconnected sets began with a pa-per delivered at an AMS meetingin April 1914 in New York andpublished later that year [15]. How-ever, his investigation was notconcerned with connected setsper se. Rather he sought to charac-terize linear continua in terms ofpoint and limit by extending a setof axioms given by F. Riesz at the

1908 International Congress of Mathematicians inRome regarding postulates enunciated by DavidHilbert in his classic book on the foundationsof geometry. Moore began by stating Lennes’sdefinition of a connected set, thereby becomingthe first person to make use of the power andgenerality of the reformulation almost nine yearsafter its initial pronouncement. Then he listedfour axioms that extended Riesz’s three, one ofwhich read, “If P is a point of S, then S − P iscomposed of two connected subsets neither ofwhich contains a limit point of the other” [15,p. 124]. This idea of expressing the complementof a set as the union of two separated, connectedsets would bear fruit in subsequent investigationsby Moore and his students.

Frigyes (Frederick in English) Riesz himselfplayed a curious role in the development of con-nected sets, having stated independently of Lennesan equivalent definition of a connected set just amonth after Lennes presented his at the December1905 AMS meeting. While Lennes’s influence wasrestricted to American mathematicians, Riesz’sdefinition seems to have remained unknown untilan investigation by R. L. Wilder on the evolution ofconnected sets seventy-two years later [16]. (See[17] for an analysis of Riesz’s contributions totopology.)

One year after Moore’s initial foray, he presenteda paper at another AMS meeting in which connectedsets were defined among axioms for the plane,leading to a paper in the Bulletin later that year [18].It is of interest to note that his first Ph.D. student,

454 Notices of the AMS Volume 56, Number 4

J. R. Kline, presented his dissertation at that April1915 meeting, continuing E. H. Moore’s policy ofinvolving students in AMS affairs early in theircareers. There is no need to examine this work ofR. L. Moore because he soon extended its majorresult appreciably in perhaps his most importantpaper, “On the foundations of plane analysis situs”[19]. This long and detailed work which we, likeMoore, abbreviate F. A., firmly solidified his repu-tation as a first-rate researcher. Furthermore, heused its sequence of theorems in his classes overthe next fifteen years to form the basis for whatwould come to be known as the Moore Method ofteaching.

The use of connected sets abounds in F. A., start-ing with their definition in the section following theintroduction and continuing throughout the paper.In one part Moore quoted Lennes’s definition ofan arc and defined a domain as a connected setof points M such that if P is a point of M , thenthere exists a region that contains P and lies inM . These definitions form the basis for Moore’sintricate proof of a major theorem [19, p. 136]:

If A and B are distinct points of adomainM , there exists an arc fromA to B that lies wholly in M .

Although the proofs of almost all 52 theorems inF. A. involve connected sets, such sets were not theprimary object of study. Rather, connectedness re-mained a tool for characterizing other kinds of sets.

The AMS role would repeat itself at an October1916 meeting where both Moore and Kline foundLennes’s formulation of connected sets to be highlyprofitable. Moore’s paper was aimed at provingthe property that any two points on a continuouscurve C in any number of dimensions form theextremities of an arc lying entirely in C. Once againhe found it necessary to supply the definition of aconnected set beforehand [20, p. 233]. This wouldnot mark Moore’s swan song with connected sets,but his student John Robert Kline (1891–1955)became more active in this regard over the nextfew years. J. R. Kline had come under R. L. Moore’sspell shortly after entering Penn in the fall of 1913and, as we have seen, presented the results ofhis dissertation at an AMS meeting in his secondyear of graduate study. He remained at Penn fortwo years after receiving his Ph.D. in June 1916.At the AMS meeting that October he proved theconverse of the following theorem on open curvesthat Moore had proved in F. A. [21, p. 178]:

If ` is an open curve in a universalset S, then S − ` = S1 ∪ S2, whereS1 and S2 are connected sets suchthat every arc from a point of S1 toa point of S2 contains at least onepoint of `.

Anna Mullikin in the 1941 yearbook fromGermantown High School.

As an indication of the extent to which connect-ed sets had become a primary tool, Kline presentedanother paper at the annual AMS meeting heldin New York just two months later. Its aim wasto generalize a result from Hausdorff’s classicGrundzüge der Mengenlehre that the complementof a countable set in n-dimensional space is alwaysconnected. Kline, like R. L. Moore before him,still felt obliged to state Lennes’s definition ofa connected set, though without attribution thistime. He did not find it necessary, however, tostate that Lennes’s formulation was equivalentto the one Hausdorff supplied in his 1914 book.Apparently Hausdorff had come upon his definitionin ignorance of Lennes.

J. R. Kline taught at Yale from 1918–19 andIllinois from 1919–20 before returning to Pennfor the rest of his life. Before leaving Penn in1918 he wrote a joint paper with R. L. Moore, theonly one that Moore ever co-authored, providingnecessary and sufficient conditions for a closedand bounded set M to be a subset of an arc interms of closed, connected subsets ofM [22]. Thusthe central structures of interest remained arcsand open curves; connected sets were relegated toauxiliary status.

Two other former students of E. H. Moore alsomade use of connected sets at this time. ArthurDunn Pitcher (1880–1923) arrived in Chicago in1907 right after Lennes had left. Pitcher receivedhis Ph.D. three years later and then embarked ona research program in Moore’s general analysis,resulting in the paper “Biextremal connected sets”,read at the same annual AMS meeting as Kline inDecember 1916. In spite of the title, connectedsets were still used only as a tool. Actually, Pitch-er had worked with Edward Wilson Chittenden(1885–1977), a 1912 Chicago Ph.D., several yearsearlier when the latter read their joint paper ata Chicago Section meeting in March 1913. Theirjoint work would ultimately be published in twopapers in the Transactions, with [23] containing

April 2009 Notices of the AMS 455

results announced at meetings in 1913 and 1916.Incidentally, A. D. Pitcher is the father of longtimeAMS secretary (1967–88) Arthur Everett Pitcher(1912–2006).

R. L. Moore must have been a very proud Dok-torvater at the April 1918 AMS meeting in NewYork. Even though he did not present a paper, hisformer student Kline delivered one characterizingsimple curves in terms of connected domains,and his second student, G. H. Hallett, announcedtheorems that appeared in his dissertation forthe Ph.D. he earned two months later. George H.Hallett Jr. (1895–1985), like Kline, wrote his thesisin geometry inspired by Moore’s early mentor G. B.Halsted. Hallett would ultimately pursue a careerin government, but before embarking on that pathhe too initiated a research program in topology, re-sulting in a paper whose sole purpose was to provethat the boundedness assumption in Lennes’scharacterization of an arc was superfluous. Whenthis result appeared in print, Hallett added, “SinceI wrote this paper it has been pointed out to me byProfessor R. L. Moore that a modification of [hisargument in F. A.] …would accomplish the sameresult” [24, p. 325]. This paper was cited as lateas 1927 by one of R. L. Moore’s initial successesat the University of Texas, Gordon Thomas Why-burn (1904–69), thus suggesting that Hallett couldhave become a successful research mathematicianshould he have chosen to remain in the field [25].

Before leaving Philadelphia, R. L. Moore men-tored a third doctoral student, Anna MargaretMullikin (1893–1975), who enrolled in the fall of1918, right after George Hallett graduated. It isnot known what motivated Mullikin to enroll atPenn but we offer a connection to Clara Bacon,who attended the October 1911 meeting whereLennes spoke about connected sets. Bacon was along-time professor of mathematics at GoucherCollege, including Mullikin’s undergraduate years,1911–15. Earlier she had pursued graduate studiesduring summer sessions at the University of Chica-go (1901–04), when she likely came in contact withfellow graduate students Lennes, Veblen, and R. L.Moore. Moreover, Bacon kept abreast of researchdevelopments, becoming the first woman to receivea Ph.D. in mathematics at Johns Hopkins in 1911.Thus it is conceivable that she knew about thesuccess of R. L. Moore at Penn and the achievementsof his students Kline and Hallett. Unfortunately wehave no firm evidence to support this contention.1

Miss Mullikin, as she came to be known, pro-gressed quickly under R. L. Moore’s special tutelageduring her first year in his class. A theorem thathad been only recently proved by W. Sierpinskiread [26]:

1We are indebted to Thomas L. Bartlow for suggesting thispossible link between Mullikin and R. L. Moore.

Mullikin nautilus.

A closed, bounded, connected setM in <n cannot be expressed as acountable union of disjoint closedsets.

Moore, well known for meticulous examinationsof axioms, challenged Miss Mullikin to discoverwhat would happen if any of the conditions in thistheorem were relaxed. By the following October, atthe start of only her second year in graduate school,she was prepared to announce her discovery atan AMS meeting in New York [27]. The publishedaccount states, “It will be shown in the presentpaper that for the case where n = 2, this theoremdoes not remain true if the stipulation that M isclosed be removed” [28, p. 144]. In summarizingher paper, F. N. Cole reported, “In one dimensionno countably infinite collection of mutually exclu-sive closed point sets ever has a connected sum[union]. One might rather naturally be inclined tobelieve that this proposition holds true also in twodimensions. Miss Mullikin shows by an examplethat this is, however, not the case” [29, p. 147].

The example, now called the Mullikin nautilus, isshown in the figure. The nautilus M is the union ofa countably infinite collection of arcs Mn, n = 1,2,…, each composed of four line segments runningfrom ( 1

2n ,0) to ( 12n ,

12n ) to (−1

2n ,1

2n ) to (−12n ,

−12n ) to

(1, −12n ). The fact thatM is connected is not obvious

in terms of the traditional geometric approach ofconnecting any two points of M while remainingwithin M . Yet it can be proved easily with theLennes definition, as follows. Assume M is theunion of two separated sets A and B. Notice thatthe restrictions An and Bn of A and B to eacharc Mn separate Mn. Since Mn is connected, eitherAn = Mn or Bn = Mn. Thus A and B are collectionsof arcs, one of which must be infinite. It is then

456 Notices of the AMS Volume 56, Number 4

easy to see that A and B must have a limit pointin common, contradicting the assumption thatthey are separated. This proves that the Mullikinnautilus M is connected, and hence serves as anexample that not every connected set is arcwiseconnected. Importantly, it illustrates the need forthe shift from the earlier constructive, geometricdefinition.

Moore arranged an instructorship for Miss Mul-likin at Texas for 1920–21 so she could completeher thesis there, and during that year two of herpapers were read at AMS meetings in New York,both dealing with connected sets. The nautilusin [27] and theorems announced in [30] and [31]formed the basis for her Ph.D. dissertation, titled“Certain theorems relating to plane connected pointsets”, which was published in the Transactionsin 1922 [28]. Miss Mullikin thus became the firstAmerican to study properties of connected setsin their own right. Her work was scrutinized in arecent paper that discusses its fifty-year mathe-matical legacy and details its role in subsequentcollaborations between the emerging schools oftopology in Poland and the U.S. [32]. Overall, then,all three R. L. Moore doctoral students at Penncontributed to the development of connected sets.

Beyond the AMSMiss Mullikin was not the first person in theworld to publish a paper in which connected setswere the primary object of study. In reviewingthe history of this topological concept, the famedmathematician/historian R. L. Wilder (1896–1982)observed that “the first paper devoted to the studyof connected sets was not published until 1921;we refer here to the classic paper Sur les ensemblesconnexes of B. Knaster and C. Kuratowski [33]”[16, p. 724]. In a strict sense, Wilder is absolutelycorrect about this fifty-page survey. Nonetheless,Fundamenta Mathematicae, the prestigious Polishjournal where [33] appeared, included four earlierpapers that established various properties of con-nected sets but received no mention in [16]. Wecite them briefly. The second paper ever publishedin Fundamenta, written by Wacław Sierpinski, ex-plored properties of connected sets that containno subsets that are continua. The other threepredecessors of [33] appeared the following year,1921. In the first, Sierpinski provided proofs ofseveral properties of connected sets; for instance,the complement of a connected set inRn containingno subsets that are continua is always connected.Fundamenta also included two short notes byStefan Mazurkiewicz solving problems posed bySierpinski, one of which established the existenceof a connected set in the plane having no connectedand bounded subsets, while the other introducedthe notion of a quasi-connected set. The latter notewas followed immediately by [33]. In addition to

these four papers, the short note by Sierpinski in1918 that motivated the Mullikin nautilus seemsalso to have escaped Wilder’s attention.

It was [33] together with [28] that elevatedconnected sets from secondary tools to primaryobjects. As we have seen, a twenty-year gestationperiod in three phases preceded this elevation,beginning with the proof of the Jordan CurveTheorem, centering on the published definition byLennes in 1911, and continuing with the work ofseveral graduates of E. H. Moore at Chicago andR. L. Moore at Penn. Connected sets did not remainthe exclusive dominion of these Moore schools inthe U.S., but they certainly accounted for the mostlasting contributions. And at the center of thisflurry of activity was the increasingly influentialAMS with its meetings in New York and Chicago andits two journals, the Bulletin and the Transactions.

AcknowledgmentsThe author is grateful to Steve Batterson and AlbertLewis for constructive criticisms on an earlierversion of this paper. He thanks Temple Universityfor the Research and Study Leave that allowed himto carry out the research for this paper. He alsothanks the Mansfield Library at the University ofMontana for permission to publish the photo of N.J. Lennes from the 1937 edition of The Sentinel.

References[1] John Kelley, General Topology, Van Nostrand, 1955.

[2] Deane Montgomery and Oswald Veblen, NelsJohann Lennes, Bull. Amer. Math. Soc. 60 (1954), 264–265.

[3] Steven Batterson, The vision, insight, and in-fluence of Oswald Veblen, Notices Amer. Math. Soc. 54(2007), 606–618.

[4] Oswald Veblen, Theory of plane curves in non-metrical analysis situs, Trans. Amer. Math. Soc. 6 (1905),83–98.

[5] , A system of axioms for geometry, Trans.Amer. Math. Soc. 5 (1904), 343–384.

[6] , The Heine-Borel theorem, Bull. Amer. Math.Soc. 10 (1904), 436–439.

[7] Thomas Hales, The Jordan curve theorem, for-mally and informally, Amer. Math. Monthly 114 (2007),882–894.

[8] Nels Lennes, Theorems on the simple finite poly-gon and polyhedron, Amer. J. Math. 33 (1911), 37–62.

[9] Oswald Veblen and Nels Lennes, Introductionto Infinitesimal Analysis; Functions of One Variable, JohnWiley, 1907.

[10] Lewis Ames, An arithmetic treatment of someproblems in analysis situs, Amer. J. Math. 27 (1905), 343–380.

[11] Nels Lennes, Curves in non-metrical analysissitus, Bull. Amer. Math. Soc. 12 (1906), 280, 284–285.

[12] , Curves in non-metrical analysis situs withan application in the calculus of variations, Amer. J. Math33 (1911), 287–326.

[13] Frank Cole, The Chicago meeting of the Ameri-can Mathematical Society, Bull. Amer. Math. Soc. 17 (1911),505–534.

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[14] , The October meeting of the AmericanMathematical Society, Bull. Amer. Math. Soc. 18 (1912),159–165.

[15] R. L. Moore, The linear continuum in terms ofpoint and limit, Annals Math. 16 (1914) 123–133.

[16] Raymond Wilder, Evolution of the topologicalconcept of “connected”, Amer. Math. Monthly 85 (1978),720–726.

[17] Wolfgang Thron, Frederic Riesz’ contributionsto the foundations of general topology, pp. 21–29 in C. E.Aull and R. Lowen, Handbook of the History of GeneralTopology, Vol. 1, Kluwer, 1997.

[18] R. L. Moore, On the linear continuum, Bull. Amer.Math. Soc. 22 (1915), 117–122.

[19] , On the foundations of plane analysis situs,Trans. Amer. Math. Soc. 17 (1916), 131–164.

[20] , A theorem concerning continuous curves,Bull. Amer. Math. Soc. 23 (1917), 233–236.

[21] J. R. Kline, The converse of the theorem concern-ing the division of a plane by an open curve, Trans. Amer.Math. Soc. 18 (1917), 177–184.

[22] R. L. Moore and J. R. Kline, On the most generalplane closed point-set through which it is possible topass a simple continuous arc, Annals Math. 20 (1919),218–233.

[23] Arthur Pitcher and Edward Chittenden, Onthe foundations of the calcul fonctionnel of Fréchet,Trans. Amer. Math. Soc. 19 (1918), 66–78.

[24] George Hallett, Concerning the definition of asimple continuous arc, Bull. Amer. Math. Soc. 25 (1919),325–326.

[25] Gordon Whyburn, Concerning continua in theplane, Trans. Amer. Math. Soc. 29 (1927), 369–400.

[26] W. Sierpinski, Un théorème sur les continues,Tôhoku Math. J. 13 (1918), 300–303.

[27] Anna Mullikin, A countable collection of mutu-ally exclusive closed point sets with connected sum, Bull.Amer. Math. Soc. 26 (1919), 146–147.

[28] , Certain theorems relating to plane con-nected point sets, Trans. Amer. Math. Soc. 24 (1922),144–162.

[29] Frank Cole, The October meeting of the Ameri-can Mathematical Society, Bull. Amer. Math. Soc. 25 (1919),145–149.

[30] Anna Mullikin, Certain theorems concerningconnected point sets, Bull. Amer. Math. Soc. 27 (1920),248–249.

[31] , A necessary and sufficient condition thatthe sum of two bounded, closed and connected point setsshould disconnect the plane, Bull. Amer. Math. Soc. 27(1921), 349.

[32] Thomas Bartlow and David Zitarelli, Who wasMiss Mullikin? Amer. Math. Monthly 116 (2009), 99–114.

[33] B. Knaster and C. Kuratowski, Sur les ensem-bles connexes, Fund. Math. 2 (1921), 206–255.

458 Notices of the AMS Volume 56, Number 4