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In algebraic topology, a branch of mathematics, the path space P X {\displaystyle PX} of a based space {\displaystyle } is the space that consists of all maps f {\displaystyle f} from the interval I = {\displaystyle I=} to X such that f = ∗ {\displaystyle f=*} , called paths. In other words, it is the mapping space from {\displaystyle } to {\displaystyle }.
The space X I {\displaystyle X^{I}} of all maps from I {\displaystyle I} to X is called the free path space of X. The path space P X {\displaystyle PX} can then be viewed as the pullback of X I → X , χ ↦ χ {\displaystyle X^{I}\to X,\,\chi \mapsto \chi } along ∗ ↪ X {\displaystyle *\hookrightarrow X}.
The natural map P X → X , χ → χ {\displaystyle PX\to X,\,\chi \to \chi } is a fibration called the path space fibration.