Poincaré conjecture
Mathematicians knew that this property was unique to the 2-sphere, in the sense that any other simply connected space that does not have edges and is small enough (in mathematician terms, that is compact) is in fact the 2-sphere. It is no longer true if we remove the idea of smallness however, as an infinitely large plane is also simply connected. Also, a regular disk (a circle and its interior) is simply connected, but it has an edge (the bounding circle).