characterization of the unit disk We know of a charcterization of spaces homeomorphic to [0,1], as being metric continua with 2 noncut points. We have as well a characterization of spaces homeomorphic to the unit circle. I can't find characterization of spaces homeomorphic to the unit disk. I'm using Willard's General Topology and Hocking and Young's Topology. Any suggestion would be appreciated.
 A: A topological characterisation of the disk is given by Zippin (see R L Wilder, "Topology of Manifolds", 1949).  Here is the statement.

Theorem [Zippin]: Suppose that $D$ is a Peano continuum, containing a subspace $C$ homeomorphic to $S^1$.  Suppose that:
  
  
*
  
*$D$ contains at least one arc spanning $C$.
  
*Every arc spanning $C$ separates $D$.
  
*No proper subset of an arc spanning $C$ separates $D$. 
  
  
  Then $D$ is homeomorphic to the unit disk, and $C$ is its boundary. 

Here an arc is an embedding of the unit interval.  Also, an arc spans $C$ if it meets $C$ exactly in its endpoints. 
A: It seems clear from the formulation that the question is about a topological characterisation of the unit disc, presumably the closed one.  The correct characterisation of the one-dimensional case  can be found in the classic "Dynamic topology" by Whyborn and Duda: a compact, connected, second countable space for which every point with the exception of two specified ones (the endpoints) is a cut point.  A perhaps not very satisfactory
characterisation of the closed unit disc is then that the space be homeomorphic to the product of two such spaces.
A: Here is an excerpt from a mathscinet review of the paper Johannes de Groot - Topological characterization of metrizable cubes (1972):

A connected $T_1$ space of dimension $n=1,2,⋯,\infty$ is homeomorphic to $I_n$ ($I_{\infty}$ being the Hilbert cube) if and only if it has a countable subbase that is both comparable and binary. A subbase $\sigma$ for the closed sets of a space is comparable if any two members of $\sigma$ disjoint from a given member of $\sigma$ are comparable (one contains the other). If every linked subcollection of $\sigma$ (i.e., each pair of the subcollection has a nonempty intersection) has a nonempty intersection, then $\sigma$ is said to be binary.

If course, $I_2$ is homeomorphic to the closed unit disk. The papers itself was fun to read. Highly recommend it!
