(When) is the range of a bijection $f\in M[G]$ in the ground model $M$?

Question

Let $M$ be a countable transitive model of ZF-P, $\mathbb P$ the set of injections from a countable subset of $\mathbb R$ into $\omega_1$ with $\le=\supseteq$. Let $G$ be a $\mathbb P^M$-generic filter over $M$, and consider the forcing extension $M[G]$. Then $f=\bigcup G\in M[G]$ is a bijection between $\mathbb R^M=\mathbb R^{M[G]}$ and $\omega_1^M=\omega_1^{M[G]}$. My questions are

(1) is the $f$-image of a closed interval in $M$, i.e., is $f([a,b]^M)\in M$?

(2) if (1) not true, is there at least a superset $S([a,b])\in M$ of $f([a,b]^M)$ such at $S(I)\cap S(J)=\emptyset$, if $I,J$ are disjoint closed intervals?

(3) if both (1), (2) are not true, is there some other forcing notion for which (1) or (2) is true?

Answer 1

No, statement (1) is not true for nontrivial closed intervals $[a,b]$. No condition can force that $f([a,b])$ is any specific subset $A\subset\omega_1$, since $A$ would have to be a proper subset, but then any condition is define on only countably many elements, and so we can always pick some $\alpha\notin A$ and extend the condition to a function mapping some $c\in[a,b]$ to $\alpha$, which would ensure $f([a,b])\neq A$.

Answer 2

In $M$, let $\omega_1= A\cup B$ be a partition into two uncountable sets.

You can (again in $M$) define a forcing notion $Q$ as the set of all 1-1 functions $p$ from a countable subset of $\mathbb R$ into $\omega_1$ such that $p(x)\in A$ for $x\in [0,1]$, and $p(x)\in B$ otherwise.

The generic function $f$ will satisfy $f"[0,1] = A\in M$.

But this definition is rather artificial. My definition of $Q$ is just a weird way to describe (something equivalent to) the forcing $\mathbb P\times \mathbb P$ (where $\mathbb P$ is your forcing).

(I admit I do not understand the motivation for your question, so my answer may not be relevant.)