I want to prove the Hausdorff property of the projective space with this definition: the sphere $S^n$ with the antipodal points identified. It's seems easy, but I can't prove formally with this definition. I take neighborhoods of the points $p$, $q$ in $S^n/\sim $, with $p\neq q$. In $S^n$ we will have four points $p$, $q$, $-p$, $-q$, with 4 disjoints neighborhoods by the Hausdorff property of $\mathbb R^{n+1}$. I don't know how to prove with rigour why when we pass to the quotient we will have two disjoints neighborhoods of $p$ and $q$.
Thanks
Best Answer
Let $q:S^n\to P^n$ be the quotient map, and let $u,v\in P^n$ with $u\ne v$; there are $x,y\in S^n$ such that $q^{-1}[\{u\}]=\{x,-x\}$ and $q^{-1}[\{v\}]=\{y,-y\}$. Let $\epsilon=\frac13\min\{\|x-y\|,\|x+y\|\}$, and set
$$U=B(x,\epsilon)\cap S^n\quad\text{and}\quad V=B(y,\epsilon)\cap S^n\;,$$
where the open balls are taken in $\Bbb R^n$. Then $U,V,-U$, and $-V$ are pairwise disjoint open nbhds of $x,y,-x$, and $-y$, respectively, in $S^n$. Moreover, $q^{-1}\big[q[U]\big]=-U\cup U$ and $q^{-1}\big[q[V]\big]=-V\cup V$. Show that $q[U]$ and $q[V]$ are disjoint open nbhds of $u$ and $v$ in $P^n$.