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Who Was Miss Mullikin?

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Thomas L. Bartlow and David E. Zitarelli

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

**2. DISSERTATION.**

Anna Mullikin received her A.B. degree in 1915 from Goucher College, where she displayed promise as a mathematician in her senior year by solving a problem on geometry in this *Monthly* [11, p. 166]. Figure 1 displays her photo from the 1915 yearbook.

For the next three years she taught mathematics at the Science Hill School (KY) and at the Mary Baldwin Seminary (VA) [A1]. She arrived in Philadelphia in the fall of 1918 to begin graduate study in mathematics at the University of Pennsylvania (Penn), where she quickly came to Moore's attention and was enrolled in his graduate class.

At that time Moore was only at the beginning of his career as a research mathematician and mentor of students. In the first year she discovered a counterexample presented below that became the starting point of her dissertation.

In her second year, 1919-1920, she continued in Moore's class and advanced her research but also returned to secondary teaching. In 1920 Moore left the University of Pennsylvania for the University of Texas. There he arranged to have Mullikin appointed as an instructor so she could complete her thesis under his guidance, perhaps the first instance of a practice he continued for twenty-five years until the University insisted that he stop.

**Figure 1.** Anna Mullikin in the 1915 yearbook *Donnybrook Fair 1915-1916*, courtesy of the Goucher College Archives. (Click to enlarge.)

Mullikin's dissertation, "Certain theorems relating to plane connected point sets," appeared in the September 1922 issue of the *Transactions of the American Mathematical Society* three months after she officially received her doctorate. The major theme, as suggested by the title, is to characterize connected sets in the plane. Sometimes, however, it is apparent that generalization to *n* dimensions is easily attainable.

Mullikin cited an important and recently proved result of the Polish topologist WacÅ?aw SierpiÅ?ski that served as the impetus for her investigation [43]:

**SierpiÅ?ski's Theorem:** A closed, bounded, connected set *M* in â?^{n} cannot be expressed as a countable union of disjoint closed sets.

Mullikin wrote: "It will be shown in the present paper that for the case where *n* = 2, this theorem does not remain true if the stipulation that *M* is closed be removed" [38, p. 144]. To accomplish this, she constructed an example of a bounded and connected set in the plane that is the disjoint union of a countable collection of closed, bounded, and connected arcs in the following way.

For each positive integer *n*, define an arc *M*_{n} to be composed of four line segments drawn from the *x*-intercept (1/(2^{n-1}), 0) to (1/(2^{n-1}), 1/(2^{n-1})), thence to (-1/(2^{n-1}), 1/(2^{n-1})) and (-1/(2^{n-1}), -1/(2^{n-1})), and finally to (1, -1/(2^{n-1})). (The abscissa of the final point is always x = 1.)

This example can be used instructively today for students in a calculus or analysis course to extend the notion of a limit point from the real line to the plane. Mullikin stated that each arc *M*_{j} contains a limit point of the union of every infinite subset of the collection {*M*_{n} : *n* â? *j*}.

Take, for example, *M*_{3}, which is composed of the line segments connecting (1/4, 0) to (1/4, 1/4) to (-1/4, 1/4) to (-1/4, -1/4) to (1, -1/4). What point on *M*_{3} is a limit point of the union of the entire collection {*M*_{n} : *n* â? 3}?

Mullikin's insightful idea was to form the countable disjoint union *M* = â?ª*M*_{n}, which we will call "Mullikin's nautilus." Students should be encouraged to sketch *M*; we reproduce Mullikin's drawing in Figure 2, which shows the arcs *M*_{n} for *n* = 1, 2, 3. The website "Demos with Positive Impact" illustrates the figure by drawing one arc after another.

Students who can identify the limit point in question are probably poised to advance from limit points in a metric space to the concept of closure in a general topological space with little difficulty, so it seems appropriate to pose another question: What is the closure of Mullikin's nautilus?

**Figure 2.** Mullikin's nautilus, Figure 1 in *Trans. Amer. Math. Soc.* **24** (1922), p. 145, printed with permission of the American Mathematical Society. (Click to enlarge.)

The set *M* is obviously bounded. Mullikin concluded that *M* is also connected by noting that each arc in the nautilus contains a limit point of every subset of *M* which consists of an infinite number of the remaining arcs. Her simple yet elegant nautilus thus provides the counterexample she sought of a bounded and connected set in the plane that is the disjoint union of a countable number of closed and bounded sets.

Moreover, it sparked several investigations afterwards. R.L. Moore modified it slightly to obtain additional results related to other theorems due to SierpiÅ?ski. (Footnote 2 in [35] cites Mullikin's role.) And a few years later, Moore's student Gordon Whyburn constructed a similar counterexample, as shown in Figure 3.

**Figure 3.** Whyburn's nautilus, Figure 3 in *Trans. Amer. Math. Soc.* **29** (1927), p. 399, printed with permission of the American Mathematical Society. (Click to enlarge.)

In typical R.L. Moore theorem-proof style, Mullikin moved directly from the nautilus to the first of five theorems in her dissertation. Her proof relied heavily on two major results from the 1911 dissertation of Zygmunt Janiszewski that was published the next year [23].

After proving Theorem 1, Mullikin moved directlyâ??and without supplying motivationâ??to her second technical result. Its proof is much shorter than the first. We reproduce the proof with original wording and notation in order to convey a sense of the Mullikin, as opposed to, say, the Bourbaki, style of presentation. Using terms adopted by Sir Michael Atiyah, we might characterize the distinction as tactical (concerned with the minutiae of the argument) as opposed to strategic (involving the development of a theory, an architectural structure) [4, p. 87].

**Theorem 2.** Let *H* be a closed and bounded set. If *H* contains disjoint closed subsets *K* and *M* but does not contain a closed, connected subset *L* with *L* â?© *K* â? â?? and *L* â?© *M* â? â??, then *H* is the disjoint union of two closed sets, of which one contains *K* and the other contains *M*.

*Proof.* There exists a positive number *Îµ* such that no point of *K* can be joined to a point of *M* by a broken line made up of intervals of length less than *Îµ* such that the end points of these intervals are points of *H*. For otherwise there would be a closed, connected "limit set" as in [the proof of] Theorem 1. This limit set would belong to *H*, since *H* is closed, and it would contain a point of *K* and a point of *M*, since *K* and *M* are both closed. This is contrary to the hypothesis.

Now let *H*_{1} denote the point set composed of *K* together with the set of all points [*P*] of *H* such that *P* can be connected with some point of *K* by a broken line of intervals of length less than *Îµ* such that the end points of these intervals belong to *H*. Let *H*_{2} denote the point set composed of all other points of *H*. *H*_{2} will contain *M* and it can easily be seen that neither *H*_{1} nor *H*_{2} contains a limit point of the other, since every point of *H*_{2} is at a distance greater than or equal to *Îµ* from every point of *H*_{1}. â??

Next, Mullikin supplied a crucial definition, stating that a set *A* *disconnects* (or *separates*) the plane if its complement â?^{2}*A* is a union of separated sets. For example, the *x*-axis disconnects the plane since its complement is the union of the upper half-plane and the lower half-plane, but none of the arcs *M*_{n} in Mullikin's nautilus does because the complement â?^{2}*M*_{n} is connected.

With this in mind, Mullikin stated and proved a powerful lemma that applies to these arcs *M*_{n} and was used in the proofs of the remaining three theorems in her dissertation.Having established these preliminaries, Mullikin was ready to prove two principal results.

With Theorem 3, Miss Mullikin achieved one of the major goals stated in the introduction to her dissertation, to prove that "a plane point set, regardless of whether it be closed or bounded, which separates its plane cannot be expressed as the sum of a countable infinity of closed, mutually exclusive point sets, no one of which separates the plane" [38, p. 144].

The next major result in Mullikin's dissertation, Theorem 4, would live in the literature for another 50 years. The proofs of Theorems 3 and 4 required elaborate constructions and a deep analysis that constituted almost two-thirds of the *Transactions* paper.

The power of the Moore Method is seen in the techniques that Mullikin discovered for these proofs. For instance, she generalized a result from Hausdorff's classic book on topology that had only appeared, and in German, a few years before [20] and she drew heavily upon three theorems from Moore's groundbreaking 1916 paper [36].

Theorem 4 would ultimately be called the Mullikin-Janiszewski Theorem, the Janiszewski-Mullikin Theorem, or simply J.M.T. after it was determined that Janiszewski had published it first, though, as we detail below, in a journal inaccessible to the Americans [22].

A footnote in the dissertation states, "Various parts of this paper were presented to the Society on October 25, 1919, December 28, 1920, and February 26, 1921" [38, p. 144]. All three meetings were held at Columbia University but Mullikin attended only the one in 1920.

Moore read her paper at the 1919 meeting. American Mathematical Society (AMS) secretary F.N. Cole reported,

In one dimension no countably infinite collection of mutually exclusive closed point sets ever has a connected sum. One might rather naturally be inclined to believe that this proposition holds true also in two dimensions. Miss Mullikin shows by an example that this is, however, not the case. [10, p. 147]

Surely, the example presented at that meeting is Mullikin's nautilus.

She must have returned home from Austin during the holiday break in 1920 to deliver her first lecture at the Society's annual meeting. The title of the lecture, "Certain theorems concerning connected point sets," almost matches the title of the dissertation.

The abstract in the secretary's report on the meeting indicates that Mullikin established Theorems 1 and 3 but it did not mention Theorem 2 or the Lemma [41, pp. 248-249]. Moreover, the presentation of Theorem 3 required that *M* be closed, a stipulation that Mullikin was able to eliminate shortly.

Theorems 4 and 5 were presented as one entity at the AMS meeting held two months later [40, p. 349]. This means that Miss Mullikin had completed her dissertation before leaving Austin.

Neither Moore nor Mullikin was present at the 1921 meeting, so her paper was read by title: "A necessary and sufficient condition that the sum of two bounded, closed and connected point sets should disconnect the plane."

Moore too submitted a paper that was read by title and dealt with issues very similar to those Mullikin presented at the annual meeting two months beforehand.

Continue to the next chapter: **3. MATHEMATICAL LEGACY.**