Poincaré conjecture
The sphere (also called the 2-sphere, as it is a 2-dimensional surface, although it is usually seen as inside a three dimensional space) has the property that any loop on it can be contracted to a point (if a rubber band is wrapped around the sphere, it is possible to slide it down to a point). Mathematicians say that the 2-sphere is simply connected. Other spaces do not have this property, for example the donut: a rubber band that goes around the whole donut once cannot be slid down to a point without it leaving the surface.