Simple unsolved math problem, 6

If you know a little point-set topology, below is an unsolved math problem whose statement is relatively simple.

Probably everyone has at least seen the Mandelbrot set in some form, as it’s a popular object of mathematical artists. Here’s a picture from Wikipedia:

wikipedia_mandelbrot-set

The formal definition is as follows. Let f_c (z)=z^2+c, where c\in \mathbb{C} is a complex number. The Mandelbrot set X is the complex plot of the set of complex numbers c for which the sequence of iterates

f_c (0), f_c (f_c (0)), f_c (f_c (f_c (0))), \dots,

remains bounded in absolute value.
We say X is locally connected if every point x\in X admits a neighborhood basis consisting entirely of open, connected sets.

Conjecture: The Mandelbrot set X is locally connected.

One thought on “Simple unsolved math problem, 6

  1. Pingback: Simple unsolved math problem, 6 | Guzman's Mathematics Weblog

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s