Difference between revisions of "Simply-connected domain"
(Importing text file) |
|||
| Line 1: | Line 1: | ||
| − | + | {{TEX|done}} | |
| + | {{MSC|55}} | ||
| − | + | 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 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 [[Homeomorphism|homeomorphic]]. See also [[Limit elements|Limit elements]] and [[Riemann theorem|Riemann mapping theorem]]. | |
| − | |||
| − | |||
| − | |||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Al}}|| K. Jänich, "Topology" , Springer (1984) pp. 148ff (Translated from German) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ma}}|| Z. Nehari, "Conformal mapping" , Dover, reprint (1975) | ||
| + | |- | ||
| + | |} | ||
Revision as of 13:01, 21 January 2014
2020 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) |
Simply-connected domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simply-connected_domain&oldid=14141