Path space (algebraic topology)

In algebraic topology, a branch of mathematics, the path space $$PX$$ of a based space $$(X, *)$$ is the space that consists of all maps $$f$$ from the interval $$I = [0, 1]$$ to X such that $$f(0) = *$$, called paths. In other words, it is the mapping space from $$(I, 0)$$ to $$(X, *)$$.

The space $$X^I$$ of all maps from $$I$$ to X (free paths or just paths) is called the free path space of X. The path space $$PX$$ can then be viewed as the pullback of $$X^I \to X, \, \chi \mapsto \chi(0)$$ along $$* \hookrightarrow X$$.

The natural map $$PX \to X, \, \chi \to \chi(1)$$ is a fibration called the path space fibration.