View source for Disc, topological

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A surface homeomorphic to a disc in a plane, i.e. an orientable [[Two-dimensional manifold|two-dimensional manifold]] of genus zero with one boundary component. A locally connected continuum without locally decomposing points is homeomorphic to a disc exactly if any simple closed curve outside a given point of this continuum decomposes it, and if there exists a simple closed curve which does not decompose it.



====Comments====
Instead of  "decomposition point"  the term cutpoint is used in Western literature: A point $x\in X$, $X$ a topological space, is a cutpoint if $X\setminus\{x\}$ is disconnected (cf. [[Connected space|Connected space]]). Also,  "separate"  is more customary then  "decompose" : A set $A\subset X$, $X$ a topological space, separates $X$ (between two points $x,y\in X$) if $X\setminus A$ is disconnected (and $x$ and $y$ are in distinct components of $X\setminus A$).

A simple closed curve is a topological image of a circle in the plane.

Another, better way to state the characterization is as follows: A locally connected metric [[Continuum|continuum]] without cutpoints is homeomorphic to a disc if and only if it contains one simple closed curve which does not separate and all other simple closed curves in it do separate it.

This theorem was proved essentially by L. Zippin in [[#References|[a2]]]. In [[#References|[a1]]] one can find a nice survey of results in this area.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.R. van Kampen,   "On some characterizations of 2-dimensional manifolds"  ''Duke Math. J.'' , '''1'''  (1935)  pp. 74–93</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Zippin,   "On continuous curves and the Jordan curve theorem"  ''Amer. J. Math.'' , '''52'''  (1930)  pp. 331–350</TD></TR></table>