A path-connected space is a topological space where any two points can be joined by a continuous path. This means there exists a continuous function from the interval [0, 1] into the space that connects the two points, allowing one to 'travel' between them without leaving the space. Path-connectedness is significant because it ensures that loops and paths can be continuously transformed into each other, which is crucial when discussing the properties of spaces, particularly in relation to the fundamental group and its calculations.
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