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 E of R^2 is said to be simply connected if both E and R^2 \E, where F\E denotes a set difference, are connected. A space S is simply connected if it is pathwise-connected and if every map from the 1-sphere to S extends continuously to a map from the 2-disk. In other words, every loop in the space is contractible.
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