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*The following article, which appeared in the *American Mathematical Monthly*, 116, no. 2 (February 2009): 99-114, is available as a PDF as well.*

Table of Contents | |

1 | Introduction |

2 | Dissertation |

3 | Mathematical Legacy |

4 | Teaching Career |

5 | Life and Family |

6 | Conclusion |

7 | Archival Sources |

8 | References |

**3. MATHEMATICAL LEGACY.**

When her instructorship at the University of Texas came to an end at the close of the academic year 1920-1921, Anna Mullikin returned to the University of Pennsylvania to complete requirements for her degree.

The public defense of her dissertation was held in January 1922, so she officially received her degree from Penn that June, one of 19 Ph.D. recipients in mathematics in the U.S. that year [8].

It took some effort to get the dissertation published in the desired outlet because the *Transactions* frowned upon the submission of dissertations. Aware of this policy, J.R. Kline asked R.L. Moore to attest that it was "a piece of work of first grade" [A2].

Moore essentially defined the qualification for the Ph.D. as doing publishable work so he happily complied at once. Upon receipt, Kline replied, "I would be very glad to read Miss Mullikin's paper and would give it prompt attention" [A3]. The dissertation was accepted soon thereafter.

We have been unable to locate any reference to the paper the following year, but its impact on researchers became dramatically apparent in 1924 with a half-dozen references. Over the next several years, the dissertation would serve as a catalyst for international cooperation and competition between the schools of topology in the U.S. and Poland.

We trace the impact of the dissertation in four distinct periods: 1924 (initiation), 1925-1929 (spread), 1929-1932 (crowning period), and 1952-1973 (modern generalizations). An account of the origin and early impact of the Moore Method briefly mentioned the influence that Miss Mullikin exerted on topology in the 1920s but it omitted many of the works discussed here [58]. Our development below fills these lacunae.

The initial period (1924) witnessed five papers by Moore and Kline, as well as one by the Polish mathematician Stefan Mazurkiewicz that included her name in the title, "Remarque sur un thÃ©orÃ¨me de M. Mullikin" [29]. After stating her Theorem 3, Mazurkiewicz wrote, "I propose to give a new proof of this remarkable theorem based on several results on closed sets" [our translation].

His one-paragraph proof relied on a recent result of Kazimierz Kuratowski and Bronislaw Knaster that had appeared in an earlier issue of the embryonic journal *Fundamenta Mathematicae* (*FM*), which had been established in 1919 to provide an outlet devoted exclusively to the theory of sets and its applications. Of course Moore and Kline were aware of Mullikin's work earlier, but Mazurkiewicz's citation suggests that the *Transactions* was becoming international.

All three Warsaw mathematicians primarily involved with founding *FM* played important roles in Mullikin's legacy. We have already mentioned Zygmunt Janiszewski (1888-1920), who had recently died in the influenza pandemic of 1918-1919. The editors of the initial volume in 1920 were Stefan Mazurkiewicz (1888-1945) and WacÅ?aw SierpiÅ?ski (1882-1969).

The second issue of *FM* from 1924 contained not only the pivotal paper by Mazurkiewicz but an important contribution by R.L. Moore that drew upon Theorem 3 once, used Theorem 4 twice, and singled out Theorem 5 [33].

This work contains a telltale footnote referring to Theorem 4 as due to Janiszewski. Moore always insisted on giving credit to the original discoverer of a theorem, although in this case he vigorously defended Mullikin's independent discovery by asserting,

A proposition which is a logical consequence of these theorems of Janiszewski's has been recently established by Miss Anna M. Mullikin in her Doctor's dissertation, which will appear soon in theTransactions... This paper had gone to the printers before either Miss Mullikin or I was aware that the proposition had already been proved. Apparently Janiszewski's paper is printed in Polish. [33, p. 190, footnote 2]

Moore had still not seen Janiszewski's paper by September 28, 1923, when he submitted his own paper to *FM*. (That date is revealed in footnote 4, p. 170, of [30]. His source for Janiszewski's results was a recent paper by the Warsaw mathematician Stefan Straszewicz in *FM*.) But once Moore was aware of Janiszewski's priority he insisted on referring to Theorem 4 as Janiszewski's Theorem, as did his later students. These citations provide an early glimpse into the evolving, and ultimately symbiotic, relationship between the schools of topology in Warsaw and Austin.

Moore also urged the reader to consult Mullikin's Theorem 5, pointing out that it was presented at an American Mathematical Society (AMS) meeting in 1921. The essential result in Moore's paper’Theorem 11, whose proof constituted almost half of it’was in the spirit of Mullikin's Theorem 3, stating that the plane contains no unbounded continuum that is a disjoint union of a countable number of continua.

Besides these two papers in *FM*, three works by Moore and another by Kline made use of Mullikin's dissertation in the *Proceedings of the National Academy of Sciences*, with Kline's following two by Moore in the May 15, 1924, issue.

Like Mullikin's investigation, Moore's two-page note [30] was inspired by the same SierpiÅ?ski work of 1918 [43]. In it, Moore proved one result, another in the spirit of Mullikin's Theorem 3: if a closed and bounded set *M* is the disjoint union of a countable number of closed and connected sets *M _{n}*, then some

Moore concluded the paper by supporting his contention that his discovery and proof were carried out independently of Mazurkiewicz, who had stated and established it that year, declaring, "I submitted my proof for publication... Sept. 28. Sometime in November I received the reprint of the article by Professor Mazurkiewicz" [30, p. 170].

Moore's second *Proceedings* article used Mullikin's Theorem 4 twice, calling it "a theorem of Janiszewski's" [31, p. 170]. In the very next paragraph, however, he relied on "a theorem of Miss Mullikin's," namely Theorem 5. In the next paper in that issue, J.R. Kline made crucial use of Theorem 4 in the proof of one of two major results he announced [26]. The remaining Moore paper from 1924 made a slight modification of Mullikin's nautilus and relied on her Theorem 4 [35].

During 1925-1929, Mullikin's theorems were used as tools in proofs in at least eleven papers by Moore, his students Wilder and Whyburn, and W.A. Wilson, who was not associated with either the Austin or Warsaw schools. Moore relied heavily on her Theorem 2 twice in [32] and again in [37], while [34] cited Theorems 4 and 5 at critical junctures in proofs.

Moore's first student at Texas, Raymond L. Wilder (1896-1982), came to Austin in 1921 after having earned a master's degree at Brown. His 1923 dissertation developed some of Mullikin's ideas and, like her, he presented his findings at more than one AMS meeting.

Wilder's third section provided a new characterization of continuous curves in any dimension, while he also drew upon a generalization of Mullikin's Theorem 1 by finding a subcontinuum *Q* of a continuum *M* in place of Mullikin's connected subset *L* of the closed, bounded and connected set *H* [50].

Mysteriously, he listed the year of her paper as 1923, a mistake repeated by several later writers. In connection with the continuing effort of Austin mathematicians to address priority issues with their Warsaw counterparts, Wilder acknowledged that, "In his article, Mazurkiewicz establishes numerous results and indicates that some of them were published in a journal to which I have not had access" [50, p. 344]. Theorem 1 became a prime component in Wilder's proofs over the next few years ([48, pp. 334 and 339], [49, p. 620], and [52, p. 351]).

Two other young mathematicians drew upon Theorem 1 at about this time. One was Gordon Whyburn (1904-1969), R.L. Moore's third doctoral student at Texas but first native Texan. His dissertation, which, like Mullikin's, was published in the *Transactions*, made critical use of Theorem 1 on four separate occasions by asserting the existence of a connected set with certain properties [47]. We already referred to this paper for its nautilus-like figure.

The first person outside the Austin-Warsaw axis to make use of Mullikin's work was Wallace Alvin Wilson (1884-1948), who used Theorem 1 to prove a fundamental lemma that was central to the proof of his own major theorem in a paper published in the *Annals of Mathematics*, the only time her dissertation was cited in that journal [53]. Wilson went on to a long career at Yale.

By the time of the crowning period, 1929-1932, Mullikin's results were so familiar that Wilder could introduce them without attribution, using the phrase "it is well known that" [51, p. 40]. But the biggest boost in this period came in the form of yet another thesis, this one by a member of the extended Moore school, Leo Zippin (1905-1995), who had won Penn's Freshman Entrance Prize in 1923.

Zippin was soon proving theorems in Kline's Foundations of Mathematics course taught in Moore-Method fashion, culminating in the dissertation, "A study of continuous curves and their relation to the Janiszewski-Mullikin Theorem." This was the first time Mullikin's name earned equal billing with Janiszewski for Theorem 4; Moore, Wilder, and Whyburn always deferred to Janiszewski.

The ongoing work of J.R. Kline's students was delivered at AMS meetings, much like Moore's, with Zippin presenting his on three occasions. The published version of his dissertation began, "In this paper... [the] principal theorems are devoted to the relation of such curves to the Janiszewski-Mullikin Theorem," and a footnote added, "The theorem is readily seen to obtain on the surface of the sphere, from the manner of its proof in the plane" [54, p. 744].

Zippin is a prime example of the close cooperation among Moore disciples at Texas and Pennsylvania. In 1929, just before Zippin received his doctorate, Kline wrote to Moore, "Zippin has applied and asked to come to Texas to work with you" [A4]. So Zippin spent the ensuing year in Austin as a National Research Fellow.

While there he began to refer to Theorem 4 as J.M.T. [56]. In two papers from 1932 Zippin continued to extend the J.M.T. to more general domains, with [55] extending to any â?* ^{n}* a result due to Clark M. Cleveland, who had obtained his Ph.D. under Moore two years earlier. Zippin's other paper suggested that Theorem 4 had become commonplace in a considerably more abstract setting: "This is a simple theorem which we have had occasion to prove for locally compact continuous curves" [57, p. 709].

Also in 1932, another academic grandson of R.L. Moore, Edwin Wilkinson Miller (b. 1905), contributed to the Mullikin legacy by making essential use of her Theorem 2. Miller had received his doctorate in 1930 under R.L. Wilder, one of the 25 Wilder produced at the University of Michigan, and had joined W.A. Wilson at Yale.

Then in 1935 the inaugural issue of the *Duke Mathematical Journal* featured a paper by E.R. van Kampen, who mentioned Mullikin tangentially by noting that characterizations of the 2-sphere lean heavily on some form of the Jordan curve theorem or the J.M.T. [46]. However, no direct mention of Mullikin would occur for another 17 years. And when it did, the centerpiece was Theorem 3, not the Janiszewski-Mullikin Theorem.

The final period of influence of Mullikin's 1922 dissertation begins in 1952 and ends in 1973 with two papers in *Fundamenta Mathematicae*, the journal that had played such a vital role in the early dissemination of her results. In 1952 W.T. van Est (Utrecht) wrote,

Recently the simplification and modernization of Miss Mullikin's proof of [Theorem 3] was proposed as a problem by Wiskundig Genootschap at Amsterdam (apparently in ignorance of Mazurkiewicz's proof). The present author succeeded in giving such a proof and at the same time generalized Miss Mullikin's theorem forndimensions (equally ignorant of Mazurkiewicz's article). [45, p. 179]

The last statement in the paper, "The proof of Miss Mullikin's theorem utilizing this homology concept remains verbally the same," indicates the extent to which recently discovered methods had been brought to bear.

Van Est's paper raised the question of finding a class of unicoherent Peano spaces in which Theorem 3 holds. It was answered in 1973 by E.D. Tymchatyn and J.H.V. Hunt (of Saskatoon) [44]. The authors of these two *FM* papers probably learned of Mullikin's work from Wilder's 1925 paper because they repeated his error of the year of publication.

Two papers in the interim show Mullikin's continuing influence. In 1963, A. Glen Haddock stated a result with a one-line proof: "This theorem follows in a straightforward manner from the Janiszvoski-Mullikin [sic] theorems" [17, p. 636].

Two years later F. Burton Jones (1910-1999), one of R.L. Moore's most influential graduates, wrote that even though the foregoing 30-year period had seen much progress on characterizations of a 2-manifold of the Jordan-curve-theorem type, "no such progress has been made on results of the second [Janiszewski-Mullikin] type. The purpose of this paper is to initiate this progress" [24, p. 497].

Having elucidated Miss Mullikin's sole publication and the durable life of her Theorems 3 and 4, we provide a fuller picture of her character by describing in the next two sections her career and some aspects of her personal life.

Continue to the next chapter: **4. TEACHING CAREER.**