Let $X$ be a compact connected metric space. Its hyperspace is denoted by $2^{X}.$ $X$ is considered as a subset of $2^{X}$ via the embedding $x\mapsto \{x\}$. Assume that $f:X\to X$ is a continuos map. Let $A$ and $B$ be two disjoint copy of $2^{X}$. We attach $A$ to $B$ along $X$, via $f$. Denote the resulting space by $2^{X}_{f}$. I Have two questions
1)How can we extract the topological invariants of $2^{X}_{f}$ in term of topological properties of $X$ and $f$?
2)Is it easy to determine (and compare) the topological spaces $2^{\mathbb{I}}_{f}$ for various polynomials $f$ which maps the interval to itself? In particular, is it true to say that $2^{\mathbb{I}}_{\text{Id}}$ is homeomorphic to $2^{\mathbb{I}}?$
