The local connectivity of a set discusses how far one must travel in the set to get to nearby points. Let A be a set in the plane and let x and y be two points in A. We can measure the distance betwen x and y in the plane. But how far apart are they in A? One way of defining their distance in A is as the smallest diamter of all the connected set in A that contain both x and y. (If you are worried about the word "smallest," then one can substitute the word "infimum" or greatest lower bound.) Let A(x,y) denote the distance in A from x to y. The "planar distance" from x to y will refer to the usual straight line distance from x to y and will have nothing to do with A.
We now pick a point z in A and ask if A is locally connected at z. We have to invent a function based on z to do this. It will be from the non-negative real numbers to the non-negative real numbers. For a real number r, let z(r) be defined as follows. Consider all points x and y in A whose planar distance is no more than r from z and record the distance A(x,y) from x to y in A. We let z(r) be the largest (or supremum or least upper bound) of all these distances A(x,y).
It might appear that as r gets close to zero, the points x and y are squeezed closer together and their distances in A will have to get close to zero as well. This is not always the case, and we say that A is locally connected at z if the function z(r) goes to zero as r goes to zero. It is now of interest to see a set that is not locally connected at some point.
In the image above, the slanted lines are
infinite in number and converge to the vertical line. The figure is
not locally connected at z since the distance in the figure from z
to infinitely many points very close to z is at least twice the
height of the vertical line. Thus the function z(r) does not go to
zero as r goes to zero.
Back to The significance of the
Mandelbrot set.
Back to The Mandelbrot Set and Julia Sets.