Dror Bar-Natan: Classes: 2000-01: Knot Theory:

Pathologies

An embedding of an interval in R3 whose complement is not simply connected:


See Hocking and Young's Topology pp. 176-177 and http://www.win.tue.nl/math/dw/personalpages/aeb/at/algtop-5.html.


See http://www.math.ohio-state.edu/~fiedorow/math655/Jordan.html.

Antoine's necklace - an embedding of a Cantor set in R3 whose complement is not simply connected:


See http://www.cs.ubc.ca/nest/imager/contributions/scharein/various/AntoinesNecklace.html.

The Alexander horned sphere - a continuous embedding of a ball in R3 whose complement is not simply connected:


See http://users.math.uni-potsdam.de/~oeitner/EIGENES/RAEUME/hornsph.htm.