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Simply Connected


SimplyConnected

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.


See also

Connected Set, Connected Space, Multiply Connected, Pathwise-Connected, Semilocally Simply Connected

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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.

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Simply Connected

Cite this as:

Weisstein, Eric W. "Simply Connected." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SimplyConnected.html

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