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''in a path-connected space''
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A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s0854201.png" /> in which all closed paths are homotopic to zero, or, in other words, a domain whose [[Fundamental group|fundamental group]] is trivial. This means that any closed path in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s0854202.png" /> can be continuously deformed into a point, remaining the whole time in the simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s0854203.png" />. The boundary of a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s0854204.png" /> may, in general, consist of an arbitrary number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s0854205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s0854206.png" />, of connected components, even in the case of simply-connected domains in Euclidean spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s0854207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s0854208.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s0854209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s08542010.png" />. 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.
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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
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* $\Gamma (0,x)= \gamma (x)$ for every $x$
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* $\Gamma (1,x) = p$ for every $x$.
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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 $q\in \Omega$.
  
See also [[Limit elements|Limit elements]].
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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 cicrle $\mathbb S^1$, 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.
  
 
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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]].
 
 
====Comments====
 
More generally, a simply-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s08542011.png" /> is a path-connected space for which each loop is contractible, i.e. whose [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s08542012.png" /> is zero for some (and hence all) base points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s08542013.png" />. The spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s08542014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s08542015.png" />, are simply connected, but the two-dimensional torus and an annulus in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085420/s08542016.png" /> are not simply connected.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Jänich,  "Topology" , Springer  (1984)  pp. 148ff  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"Z. Nehari,  "Conformal mapping" , Dover, reprint  (1975) pp. 2</TD></TR></table>
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{|
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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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Revision as of 13:01, 21 January 2014

2010 Mathematics Subject Classification: Primary: 55-XX [MSN][ZBL]

The term refers often to open subsets $\Omega$ (which usually are assumed to be 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 $\pi_1 (\Omega)$ 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 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 cicrle $\mathbb S^1$, 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 homeomorphic. See also Limit elements and Riemann mapping theorem.

References

[Al] K. Jänich, "Topology" , Springer (1984) pp. 148ff (Translated from German)
[Ma] Z. Nehari, "Conformal mapping" , Dover, reprint (1975)
How to Cite This Entry:
Simply-connected domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simply-connected_domain&oldid=14141
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article