A pathwise-connected domain is said to be simply connected (also called 1-connected) if any simple closed curve can be shrunk to a point continuously in the set. If the domain is connected but not simply, it is said to be multiply connected. In particular, a bounded subset  of  is said to be simply connected if both  and , where  denotes a set difference, are connected.

A space  is simply connected if it is pathwise-connected and if every map from the 1-sphere to  extends continuously to a map from the 2-disk. In other words, every loop in the space is contractible.

REFERENCES:

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991.

Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 27, 1999.