Let $X$ be a finite pointed CW complex (where the basepoint forms a $0$-cell) and $A$ be a CW subcomplex of $X$ containing its basepoint.
I am trying to determine if $X/A$ is also a finite pointed CW complex, and have had some issues with checking the Hausdorff property. I am aware that in many cases the quotient of a Hausdorff space is not necessarily Hausdorff.
It is clear that if $[x]$, $[y]$ are equivalence classes with a single point outside of $A$, then there exist disjoint $U$, $V$ open in $X/A$ with $[x]\in U$ and $[y]\in V$. With the remaining case where $[x]\neq A$ and $[y]=A$, it looks like I need to find disjoint $U,V$ open in $X$ with $x\in U$, $0_X\in V$ and $A\subseteq V$.
We have that $x\in e$ for some cell $e$ disjoint from $A$ and can use the attaching map $\Phi_e:D^n\to X$ to obtain $U$ open in $X$ with $U\cap e$ corresponding to a subset of an open ball centred at $x$ in $D^n$. I am puzzled about this because the closure of a cell outside of $A$ could include points inside $A$. How I can I prove that $X/A$ is Hausdorff? Have I missed something?