title: Path space (algebraic topology)
text: # 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.
  
  