Extending a homeomorphism

Question

Let $A\subset B$ be a closed nonempty subspace of $B$ and let $C\subset D$ be a closed nonempty subspace of $D$. All spaces are Hausdorff.

Suppose that we have a homeomorphism $f:B-A\rightarrow D-C $ and suppose that $\overline{B-A}=B/A$ and $ \overline{D-C}=D/C$. Can we extend $f$ to a homeomorphism

$f^{ext}: B/A\rightarrow D/C$ ?

where $B/A$ and $D/C$ are quotient space and $B-A$ (resp. $D-C$) is seen as a subspace of $B/A$ (resp. $D/C$) in a canonical way.

Answer 1

No, you can't.

Take $B=D=\mathbb{C}$, and $A=C=\{0\}$, so $B/A=B$, $D/C=D$ naturally. The map $f(z)=1/z$ is a homeomorphism $\mathbb{C}-\{0\}\to\mathbb{C}-\{0\}$, but it does not extend to $\mathbb{C}\to\mathbb{C}$ because you are "filling in the wrong point".