In algebraic topology, a branch of mathematics, the based path space PX of a pointed space (X, *) is the space that consists of all maps f from the interval I = 0, 1 to X such that f(0) = * , called based paths. In other
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In algebraic topology, a branch of mathematics, the based path space PX of a pointed space (X, *) is the space that consists of all maps f from the interval I = 0, 1 to X such that f(0) = * , called based paths. In other words, it is the mapping space from (I, 0) to (X, *) . A space X^I of all maps from I to X, with no distinguished point for the start of the paths, is called the free path space of X. The maps from I to X are called free paths. The path space PX is then the pullback of X^I \to X, \, \chi \mapsto \chi(0) along * \hookrightarrow X .
