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.
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.