 We then develop a language and theory for stable paths within this filtration , inspired by ideas of persistent homology .
<meaning-changed>  We then develop a language and theory for stable paths within this filtration , inspired by ideas of persistent homology .
 We then develop a language and theory for stable paths within this filtration . Unlike standard results on stability in topological persistence, our definition of path stability aligns exactly with the above result on stability of cover filtration .