That would certainly be "strange and creative"! That's exactly the kind of thing I wanted to see. But, I have a few problems:

A) I know so little about what you just said that I only

*think* you are talking about fractals (I know who Benoit Mandelbrot is)

B) I think you're idea involves discussing a sequence of connected sets

such that

. What does convergence mean for a sequence of sets? Secondly if I take a heuristic idea of what that would mean a reasonable counterexample to a sequence of connected sets converging to a connected sets would be

since each

is connected but

which is not. Does that maybe capture an idea of converging of a sequence of sets?