# Simply-connected domain

*in a path-connected space*

A domain in which all closed paths are homotopic to zero, or, in other words, a domain whose fundamental group is trivial. This means that any closed path in can be continuously deformed into a point, remaining the whole time in the simply-connected domain . The boundary of a simply-connected domain may, in general, consist of an arbitrary number , , of connected components, even in the case of simply-connected domains in Euclidean spaces , , or , . The boundary of a bounded planar simply-connected domain consists of a single connected component; all planar simply-connected domains are homeomorphic to each other.

See also Limit elements.

#### Comments

More generally, a simply-connected space is a path-connected space for which each loop is contractible, i.e. whose fundamental group is zero for some (and hence all) base points . The spheres , , are simply connected, but the two-dimensional torus and an annulus in are not simply connected.

#### References

[a1] | K. Jänich, "Topology" , Springer (1984) pp. 148ff (Translated from German) |

[a2] | Z. Nehari, "Conformal mapping" , Dover, reprint (1975) pp. 2 |

**How to Cite This Entry:**

Simply-connected domain.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Simply-connected_domain&oldid=14141