In class, I was given a constructive definition of a CW-complex and told that it's an easy exercise to prove that it is Hausdorff. I included the definition given in class below;
We set $X^0 = \bigsqcup_{i\in I_0}D^0_i$. That is, $X^0$ is the union of disjoint 0-balls (i.e. points). Now, for each $n \in \mathbb{N}$, if $X^{n-1}$ is given then we define attaching maps for each $i$ in some index set $I_n$; $$ f_i^n:\partial D_i^n \to X^{n-1} $$ Then $$ X^n = \left.\left(X^{n-1}\sqcup \bigsqcup_{i\in I_n}B_i^n\right)\middle/ \left\{p\sim f_i^n(p) : p\in \partial D_i^n, i\in I_n\right\}\right. $$
Being very new to algebraic topology, I do not see how the proof is obvious. The statement seems clear, but I am having a hard time using the definition to construct a rigorous proof. If anyone could help me prove this (and maybe elaborate on where detail is needed) I would appreciate it.
Thanks in advance!