6h: Is the Mandelbrot set connected?
Description
This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by others.
6h: Is the Mandelbrot set connected?
The Mandelbrot set is simply connected. This follows from a theorem
of Douady and Hubbard that there is a conformal isomorphism from the
complement of the Mandelbrot set to the complement of the unit disk. (In
other words, all equipotential curves are simple closed curves.) It is
conjectured that the Mandelbrot set is locally connected, and thus pathwise
connected, but this is currently unproved.
Connectedness definitions:
Connected: X is connected if there are no proper closed subsets A and B of
X such that A union B = X, but A intersect B is empty. I.e. X is connected
if it is a single piece.
Simply connected: X is simply connected if it is connected and every closed
curve in X can be deformed in X to some constant closed curve. I.e. X is
simply connected if it has no holes.
Locally connected: X is locally connected if for every point p in X, for
every open set U containing p, there is an open set V containing p and
contained in the connected component of p in U. I.e. X is locally connected
if every connected component of every open subset is open in X.
Arcwise (or path) connected: X is arcwise connected if every two points in
X are joined by an arc in X.
(The definitions are from _Encyclopedic Dictionary of Mathematics_.)
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