Theorem 4.1

  is connected.

Proof

First we show that for ,

is connected. The idea is to apply Lemma 4.1 to the function f which assigns to the set of zeros of

inside . To make a continuous map of this requires some manipulation.

By Jensen's theorem, as was shown in Section 2, there is an upper bound n on the number of zeros that a power series with 0,1 coefficients can have inside . Let M be with the annulus shrunk to a point P. (Therefore M is topologically a sphere.) To each power series , , we assign the set of zeros inside , (counted with multiplicities) and throw in extra copies of the point P as necessary to bring the total number of points to n. Since the order of these n elements of M is unspecified, we obtain a point of . Let be this point.

We claim that this map

is continuous. This follows easily from Rouché's theorem; if two power series agree in the first m coordinates for m sufficiently large then their zeros inside will be within . Some may escape or enter the disk, but this is not a problem, since in the topology on M, P is close to all points z with |z| sufficiently near .

We next check condition of Lemma 4.1. This is easily done using the following trick: given , let . Then , , and (we extend v,w to infinite vectors by appending 0's), since

and

have the same zeros inside . Therefore we may apply Lemma 4.1 and deduce that the image of f is path connected.

Since , we may apply Lemma 4.2 with to deduce that with the annulus shrunk to a point P is a connected subset of M. This is equivalent to the connectivity of .

Since is the decreasing intersection of the compact connected sets , it too is connected. So is its image under . Finally, is the union of these two sets, which meet on the unit circle, so is connected as well.