The path-lifting property is a crucial concept in algebraic topology that states that given a fibration and a continuous path in the base space, there exists a unique path in the total space starting at a specified point that projects down to the given path. This property ensures that paths can be 'lifted' to the total space while maintaining their continuity and endpoint conditions. It is essential for understanding how spaces relate through fibrations and facilitates the construction of long exact sequences associated with these fibrations.
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