# 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.