Simply-connected domain

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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.


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.


[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:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article