stylistic:style_transfer, structural:summarization, linguistic:paraphrasing
Basic properties

The Mandelbrot set is a compact set, since it is closed and contained in the closed disk of radius 2 around the origin. A point c {\displaystyle c} belongs to the Mandelbrot set if and only if | z n | ≤ 2 {\displaystyle |z_{n}|\leq 2} for all n ≥ 0 {\displaystyle n\geq 0}. In other words, the absolute value of z n {\displaystyle z_{n}} must remain at or below 2 for c {\displaystyle c} to be in the Mandelbrot set, M {\displaystyle M}, and if that absolute value exceeds 2, the sequence will escape to infinity. Since c = z 1 {\displaystyle c=z_{1}}, it follows that | c | ≤ 2 {\displaystyle |c|\leq 2}, establishing that c {\displaystyle c} will always be in the closed disk of radius 2 around the origin.

The intersection of M {\displaystyle M} with the real axis is the interval {\displaystyle \left}. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family,

The correspondence is given by

This gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set.

Douady and Hubbard showed that the Mandelbrot set is connected. They constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk. Mandelbrot had originally conjectured that the Mandelbrot set is disconnected. This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of M {\displaystyle M}. Upon further experiments, he revised his conjecture, deciding that M {\displaystyle M} should be connected. A topological proof of the connectedness was discovered in 2001 by Jeremy Kahn.

The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of M {\displaystyle M}, gives rise to external rays of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle.

The boundary of the Mandelbrot set is the bifurcation locus of the family of quadratic polynomials. In other words, the boundary of the Mandelbrot set is the set of all parameters c {\displaystyle c} for which the dynamics of the quadratic map z n = z n − 1 2 + c {\displaystyle z_{n}=z_{n-1}^{2}+c} exhibits sensitive dependence on c, {\displaystyle c,} i.e. changes abruptly under arbitrarily small changes of c. {\displaystyle c.} It can be constructed as the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates. The Mandelbrot curves are defined by setting p 0 = z, p n + 1 = p n 2 + z {\displaystyle p_{0}=z,\ p_{n+1}=p_{n}^{2}+z}, and then interpreting the set of points | p n (z) | = 2 {\displaystyle |p_{n}(z)|=2} in the complex plane as a curve in the real Cartesian plane of degree 2 n + 1 {\displaystyle 2^{n+1}} in x and y. Each curve n > 0 {\displaystyle n>0} is the mapping of an initial circle of radius 2 under p n {\displaystyle p_{n}}. These algebraic curves appear in images of the Mandelbrot set computed using the "escape time algorithm" mentioned below.