## Basic Properties of the Mandelbrot Set

The Mandlebrot set a) is a compact subset of b) is symmetric under reflection in the real axis

c) meets the real axis in the interval d) has no holes it it

Proof

For each iterate of 0, defines a polynomial function of    and so on. In general for To prove a) define the set for n=1,2,3,... so that and so on. and so that M is bounded .

Each set is closed because it has complement which is open. In fact if for some then this inequality must also hold for all in some open disc with centre by the continuity of the function It follows that must be closed because if then for some and so some open disc with centre must lie outside and hence outside This proves that is closed and bounded, or compact.

To prove b), notice that because is a polynomial in with real coefficients, for but if and only if by symmetry of in the real axis.

To prove c), note is disconnected for and so for and To prove d) we can prove is connected so that each pair of points in can be joined by a path in Consider the set We can show that each point in can be joined to a point of by a path in Suppose in fact that is a point of which cannot be joined to in this way. Define  since and is open because if can be joined to then so can any point of any open disc in with centre and is connected because pairs of points in can be joined in via thus is a subregion of Since cannot be joined in to and is open we deduce Now use the Maximum Principle. If then since if it were we could enlarge slightly. Thus for By applying the Maximum Principle to each polynomial function on we obtain for and contradicting that so is connected. 