I have the following definition:
Let $D,T \subseteq\mathbb{R}^2$ be the closures of connected open subsets of $\mathbb{R}^2$, i.e. $D=\overline{A}$ and $T=\overline{B}$ with $A,B$ connected open subsets of $\mathbb{R}^2$. A map $\Phi:T\to D$ is said to be an admissible change of parameters if $\Phi$ is bijective, $C^1$ with inverse $C^1$, and such that det $Jac(\Phi)\ne 0$ in $\mathring{T}$ with $\mathring{T}$ be the interior of $T$.
Question 1) Is $\mathring{T}=B$ ?
Question 2) Since $\Phi^{-1} \circ \Phi=id$ then we have $Jac(\Phi^{-1})_{\Phi(x)}\circ Jac(\Phi)_x=I_2$ (identity matrix of order $2$) for each $x \in T$ so is the expression "...and such that det $Jac(\Phi)\ne 0$ in $\mathring{T}$" superfluous in the above definition?
Question 3) The map $\Phi$ maps $\mathring{T}$ in $\mathring{B}$?
Question 4) The map $\Phi$ is an homeomorphism. In general homeomorphisms preseve interior points? More precisely: If $X,Y$ are topological spaces, $S\subseteq X$ and $W\subseteq Y$ are subspaces, and $f:S \to W$ is an homeomorphism, does $x \in \mathring{S}$ implies $f(x) \in \mathring{W}$? (Where $\mathring S$ is the interior of $S$ in $X$ and $\mathring W$ is the interior of $W$ in $Y$).
7The interior of $T$ is not always $B$. Suppose that $B$ is an open ball with its center removed, its adherence is a closed ball $T$.
Suppose $B$ is an open ball, $A$ is $B$ with its center removed, $T=D$ and let $\Phi$ be the identity of $T$, $\Phi$ does not map $B$ to $A$.