# Multiply-connected domain

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in a path-connected space

A domain $D$ in which there are closed paths not homotopic to zero, or, in other words, whose fundamental group is not trivial. This means that there are closed paths in $D$ which cannot be continuously deformed to a point while remaining throughout within $D$, or, otherwise, a multiply-connected domain $D$ is a domain which is not a simply-connected domain.

The order of connectivity of a plane domain $D$ in $\mathbf R ^ {2}$ or $\mathbf C = \mathbf C ^ {1}$( or in the compactification of these spaces, $\overline{\mathbf R}\; {} ^ {2}$ or $\overline{\mathbf C}\;$) is the number of (homologically) independent one-dimensional cycles, that is, the one-dimensional Betti number $p ^ {1}$ of $D$. If the number $k$ of connected components of the boundary of a plane domain $D$, considered as a domain in the compactified space $\overline{\mathbf R}\; {} ^ {2}$ or $\overline{\mathbf C}\;$, is finite, then $p ^ {1} = k$; otherwise one sets $p ^ {1} = \infty$. When $p ^ {1} = 1$, $D$ is a simply-connected domain, when $p ^ {1} < \infty$ it is a finitely-connected domain (one also uses such terms as doubly-connected domain, triply-connected domain $\dots$ $k$- connected domain), when $p ^ {1} = \infty$, $D$ is an infinitely-connected domain. All plane finitely-connected domains with equal order of connectivity, $k$, are homeomorphic to each other. By removing from such a domain $D$ all the points of $k - 1$ cuts, that is, Jordan arcs joining pairs of connected components of the boundary, it is always possible to obtain a simply-connected domain $D ^ {*} \subset D$. About the conformal types of plane multiply-connected domains see Riemann surfaces, conformal classes of.

The topological types of domains in $\mathbf R ^ {n}$, $n \geq 3$, or $\mathbf C ^ {m}$, $m \geq 2$, are far more diverse and cannot be characterized by a single number. Here, sometimes, the term "multiply-connected domain" (with various provisos) is used when the fundamental group is trivial but some higher-dimensional homology group is not trivial.

On the other hand, in (algebraic) topology one defines an $n$- connected space as a space $X$ such that any mapping from a sphere $S ^ {m}$, $m \leq n$, into $X$ is homotopic to zero. Thus, $0$- connectedness is the same as path connectedness.