This question is quite specific, but it may admit answers in more general contexts.

Consider a subset $\Lambda \subset D^2$ where $D^2$ is the two dimensional disk.

We consider in $\Lambda$ an equivalence relation such that the equivalence class of each point is a contractible compact set.

Assume that the quotient map $p: \Lambda \to \Lambda / \sim$ which is continuous has as image a cantor set.

The question is: If we extend the equivalence relation to the whole disk by considering for each point $x\in D^2 \backslash \Lambda$ the equivalence class is the singleton $\{x\}$ do we have that the proyection to the quotient of the whole disk is homeomorphic to the disk?

If the answer is negative, can we ask more to the equivalence classes in $\Lambda$ in order to have the result?

Maybe the question is trivial or well known, but I could not find either a reference nor an answer by myself.

EDIT: In view of Franklin's answer. I am supposing that $\Lambda$ is contained in the interior of the disk (which I am assuming closed).