# 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:

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.