Limit Points and Connected Sets in the Plane - Connected Sets

In the official report from the AMS meeting where Mullikin’s paper was read, longtime Society secretary Frank 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” [1, p. 147].  Mullikin expressed her motivation somewhat differently in the introduction to her dissertation.  After citing Lennes’ formulation of a connected set, she stated Sierpinski’s Theorem that a closed, bounded, and connected set in n dimensions cannot be written as a disjoint union of a countable number of closed sets.  Now, each arc in the Mullikin nautilus M = U M_n is obviously closed, and the set M is clearly bounded.  Mullikin asserted that M is also connected, which thus achieved her goal.  She wrote, “It will be shown in the present paper that for the case where n = 2, this theorem [of Sierpinski] does not remain true if the stipulation that M is closed be removed” [3, 144].

Mullikin’s Theorem: The nautilus  is connected.

Proof.  Suppose M is the union of separated sets A and B.  Then each arc M_n = A_n U B_n, where A_n=A \(\cap\) M_n and B_n=B \(\cap\) M_n.  But M_n is connected, so either  M_n = A_n  or M_n = B_n .  Therefore A and B are collections of arcs.  One of these collections must be an infinite set, say A.  By Mullikin’s Lemma, if B consists of at least one arc then it will contain a limit point of A, contradicting the assumption that A and B are separated.  Consequently B = \(\emptyset\), hence M is connected.

Armed with an intuitive feel for limit points of the nautilus, the fact that M is connected can be underscored by viewing a QuickTime movie whose dynamics reinforce Mullikin’s Theorem.  Click here to view it.  A module for presenting this in the classroom can be obtained at the Math Gateway partner “Demos with Positive Impact” by clicking  here.