An embedding of an interval in * R^{3}* 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
* R^{3}* 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 * R^{3}* whose complement is not simply
connected:

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