Simply-connected domain

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 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 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)