View source for Simply-connected domain

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The term refers often to open subsets $\Omega$ (which usually are assumed to be [[Connected space|connected]]) of the Euclidean space $\mathbb R^n$ where each closed path is homotopic to zero. A closed path, namely a continuous map $\gamma : \mathbb S^1 \to \Omega$, is ''homotopic to zero'' (or ''contractible'') if it can be deformed continuously to a point, i.e. if there is a continuous map $\Gamma: [0,1]\times \mathbb S^1\to \Omega$ and an element $p\in \Omega$ such that 
* $\Gamma (0,x)= \gamma (x)$ for every $x$ 
* $\Gamma (1,x) = p$ for every $x$.
In other words, the [[Fundamental group|fundamental group]] $\pi_1 (\Omega)$ of $\Omega$ is trivial. Note that the connectedness assumptions guarantees that, if $\gamma$ can be deformed to a point $p\in \Omega$, then it can also be deformed to ''any other'' point $q\in \Omega$.

More in general the same concept and definitions apply literally to any [[Path-connected space|path-connected]] [[Topological space|topological]] space $X$ and to any path-connected subset of $X$. The spheres $\mathbb S^n$, with $n\geq 2$, are simply connected, whereas the circle $\mathbb S^1$, the $n$-dimensional tori $\underbrace{\mathbb S^1 \times \ldots \times \mathbb S^1}_n$ and the annuli $\{x\in \mathbb R^2 : r<|x|<R\}$ are not simply connected.

The boundary of a simply-connected open domain $\Omega$ may, in general, consist of an arbitrary number (even infinite) of connected components, even in the case of simply-connected domains in the Euclidean space $\mathbb R^n$ ($n\geq 2$). However, if $\Omega\subset \mathbb R^2$ is, in addition to simply-connected, also bounded, then its boundary is connected. All planar simply-connected domains are [[Homeomorphism|homeomorphic]]. See also [[Limit elements|Limit elements]] and [[Riemann theorem|Riemann mapping theorem]].

====References====
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|valign="top"|{{Ref|Al}}||  K. Jänich,   "Topology" , Springer  (1984)  pp. 148ff  (Translated from German)
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|valign="top"|{{Ref|Ma}}|| Z. Nehari,   "Conformal mapping" , Dover, reprint  (1975)
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