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 , where is a complex number. The Mandelbrot set is the complex plot of the set of complex numbers for which the sequence of iterates $latex f_c(0)$, $latex f_c (f_c (0))$, $latex f_c (f_c (f_c (0)))$, etc., remains bounded in absolute value.
Conjecture: The Mandelbrot set is locally connected.
We say is locally connected if every point admits a neighborhood basis consisting entirely of open, connected sets. If you know a little point-set topology, this is a conjecture whose statement is relatively simple.