Limit groups with respect to Thompson's group $F$ and other hereditarily separating groups

We prove that a limit group over Thompson's group $F$ cannot be an HNN-extension of $F$ with respect to a finitely generated subgroup. On the other hand we give an example of an $F$-limit group which is a centralized HNN-extenstions of $F$. We also study relatively limit groups with respect to $F$ and some other known groups of homeomorphisms of topological spaces. This paper revises and extends some results of the second author from arXiv:1308.6330.