Let $X,d$ be a set with some additional algebraic or topological structure.
Let $f:X\to X$ be a function into a proper subset of $X$ such that the restriction of $f$ to the bijection $X_n\to X_{n+1}$, where $X_{n+1}$ is a proper subset of $X_n$ preserves the topological structure of $X$, i.e. it is a homeomorphism.
Now let the sequence of homeomorphisms be well-founded under composition. For example, $\Bbb Z_2\mapsto2\Bbb Z_2$ has this well-founded property, whereas $\Bbb Q_2\mapsto2\Bbb Q_2$ does not.
$f$ cannot therefore be a homeomorphism on $X$ that preserves the structure. However as a function from $f:X\to\mathrm{Im}_f(X)$ it is a homeomorphism.
Do such maps, and infinite sequences of homeomorphic sets (i.e. well-founded but otherwise structure-preserving) have names, and if so, what are they called?
In case it's related, the case I'm studying has the additional property that the infinite limit of the homeomorphism sequence $\lim_{n\to\infty} x\mapsto f^n(x)$ is a surjection over $X$. (However, it no longer injects.) I want to know more about that, so any pointers would be appreciated. I suspect this might imply $X$ is totally disconnected compact space.