Some remarks on connected sets
P. Erdös · Bulletin of the American Mathematical Society · 1944
This note will consist of a few disconnected remarks on connected sets.i. Swingle 1 raised the question whether the plane is the sum of c disjoint biconnected sets.The answer as we shall show is affirmative.First we construct a biconnected set A with a dispersion point x such that any two points of A -x can be separated.(The first such set was constructed by Wilder. 2 Our construction will be very similar to that of Burton Jones. 3 ) Our biconnected set will contain the origin O, at most one other point on lines through the origin with irrational slopes, and no point other than the origin on lines with rational slopes.Further it will contain at least one point on every cut of the plane.It is easy to see that such a set exists.Every cut of the plane contains a closed subset which also cuts the plane, and the power of closed sets is known to be c.Let us well order the closed cuts G, C2, • • • , C y , • • • , y<0«, where 0« is the least ordinal number of power c.We construct A as follows: O belongs to A, We shall choose a point x y on C y and we shall have A = \Jx y .We shall determine x y by transfinite induction.Suppose we have already determined #a, S<7, we determine x y as follows: if C y contains O then x y = D.If C y does not contain O then clearly C y has to intersect c lines through £).Therefore we can find a point Xy(ECy such that (O, x y ) has irrational slope and does not go through any other point x*, 5<y.(We denote by (a, b) the line through the points a and b.)This way we construct A, Clearly A is biconnected.First of all A is connected since it intersects every cut of the plane.Also any two points of A -£5 can be separated since the two points xi and #2, say, are on different irrational lines through the origin, and a rational line, which of course does not intersect A -£), will separate them.This completes the proof.The origin we call the center of A.Now we shall split the plane into the sum of c such disjoint sets Ay, y<£lat fla the smallest ordinal of power c.Let us well order the points Xy of the plane and the closed cuts C y of the plane.Our first step is to select X\ as the center of A\ and a suitable point of A\ on &.