Path-connected space

A path-connected space is a type of mathematical space where any two points can be linked by a continuous path within that space. Think of it as a landscape where you can travel from any location to another without leaving the space or having to jump over gaps. For example, a solid disk or a connected piece of a rubber sheet are path-connected, because you can draw a smooth line inside them between any two points without lifting your pencil. If such a path exists for every pair of points, the space is considered path-connected.