Limit Points and Connected Sets in the Plane - Limit Points

In order to answer the Fundamental Question, it is necessary to formulate a definition of connected sets that captures the intuitive notion of continuity in a rigorous and applicable manner.  To accomplish this we need two preliminary definitions.  First, the closure of a set X, denoted by \(\overline{X}\), consists of X and all of its limit points.  Then we say that two distinct sets A and B are separated if neither contains a limit point of the other, that is, if \(A \cap \overline{B} = \emptyset\) and \(\overline{A} \cap B = \emptyset\).  For instance, the upper half-plane and lower half-plane are separated.  We are now poised to state the definition of connected sets first proposed by the Norwegian-born American mathematician Nels J. Lennes [2].

FUNDAMENTAL DEFINITION: A set of points is connected if it cannot be written as the union of two separated sets.

At heart, this definition expresses connectedness in terms of limit points, so to gain insight into Mullikin’s nautilus we restate an assertion Miss Mullikin made.

MULLIKIN’S LEMMA: Each arc M_j contains a limit point of every subset of M which consists of an infinite number of the remaining arcs.

In Mullikin’s Lemma, the intercept x_j = (1/2^j-1, 0) is a point on the arc  M_j  that is a limit point of every subset of M consisting of an infinite number of the remaining arcs.  Click here to see a QuickTime movie demonstrating the convergence geometrically for x₃ = (1/4, 0).

An understanding of this movie permits one to form the closure `M of Mullikin’s nautilus M = U M<sub>n</sub>.  For if  p = (x, 0) with \(0 \leq x \leq 1\), then p Î`M.    Hence `M = M U I, where I is the unit interval in the plane.  Click here to see a QuickTime movie demonstrating this convergence for x = 1/3.  Since (1/3, 0) is in\(\overline{M}\)  but not in M, it follows that M is not closed.