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). 



Collapsing contractible subsets of the two-disk.
I think you may find the *Bing shrinking criterion* useful. 


First, assume $\Lambda$ itself be closed (hence compact) in $D$. More generally, equivalence classes can form a so-called upper semi-continuous decomposition of your compact initial space $X$, namely one such that $X/\sim$ is Hausdorff (necessary anyway), making the quotient map $p$ closed.


Bing shrinking criterion : $p:X\to X/\sim$ is a uniform limit of homeomorphisms iff for any $\epsilon>0$, there is an homeomorphism $h\_\epsilon$ of $X$ that send any equivalence class into a set of diameter $<\epsilon$ which is moreover contained in the $\epsilon$-neighborhood of the original class.


And one proof (by Robert Edwards) is a beautifully simple application of Baire's theorem, which works for any usc decomposition of a compact metric space. See [this 1979 Bourbaki talk](http://www.numdam.org/item?id=SB_1978-1979__21__278_0) 
(page 10) by Edwards himself.


