# Hausdorff quotient collapsing and separating a prescribed collection of disjoint closed subsets

Let $$X$$ be a compact Hausdorff space (I don't mind assuming it's metrizable). Let $$A_i$$ $$i\in \mathbb{N}$$ be a collection of disjoint closed subsets of $$X$$.

My question: Does there exist a Hausdorff quotient $$Y$$ such that the quotient $$\phi: X \to Y$$ satisfies:

a)$$\phi(A_i)$$ is a single point for each $$i$$.

b)For distinct $$i$$ and $$j$$, $$\phi(A_i)\neq \phi(A_j)$$.

Remark: The "obvious" equivalence relation obtained by collapsing each of the $$A_i$$ does not necessarily give a Hausdorff quotient, since a sequence of points $$a_i \in A_i$$ can accumulate to a point outside the union of the $$A_i$$. Also the closure of this relation does not appear to be transitive.

No. Consider $$Y=\{0,n^{-1}\mid n\in \mathbf N\setminus \{0\}\}$$, $$X=Y\times \{-1,1\}$$ and $$A_n=\{(n-1)^{-1}\}\times \{-1,1\}$$ for $$n>1$$, $$A_0=\{(0,-1)\}$$ and $$A_1=\{(0,1)\}$$.

Since $$X=\bigcup_n A_n$$, there is only one quotient such that images of $$A_n$$ are disjoint singletons, namely a convergent sequence with two limits. This quotient is not Hausdorff.