MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 2 added 121 characters in body

Have you tried solenoids? Solenoids certainly seem like they would satisfy your definition. Also, Antoine's necklace is another likely candidate.

Edit: Sorry, I didn't realize you required it to be connected. Solenoids still work, but Antoine's necklace is out.

show/hide this revision's text 1 [made Community Wiki]

Have you tried solenoids? Solenoids certainly seem like they would satisfy your definition. Also, Antoine's necklace is another likely candidate.