A space $X$ whose topology agrees with the final topology with respect to all maps $I\to X$ is often called a *[delta-generated][1]* ($\Delta$-generated) space. The category of $\Delta$-generated spaces is a convenient category of spaces (in the sense of Steenrod) and is used in Diffeology, String Topology, and generalized covering spaces theories.


  [1]: https://ncatlab.org/nlab/show/Delta-generated+space