A path-connected space is a topological space where any two points can be connected by a continuous path within that space. This means that for any pair of points, there exists a continuous function mapping the interval [0, 1] to the space, indicating a way to 'travel' from one point to another without leaving the space. Path-connectedness is an important property when discussing covering groups and the fundamental group, as it relates to the ability to define loops and continuous transformations.
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