Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / E / 2 (E2)

A Hausdorff space is one in which any distinct points $x,y$ have disjoint open neighborhoods: there exist open sets $U,V$ with $x\in U$, $y\in V$ and $U\cap V=\varnothing$.

A map $f:X\to Y$ is continuous if the inverse image of every open subset of $Y$ is open in $X$:

For the natural projection $p:X\to X/R$, the quotient topology declares $U\subseteq X/R$ open exactly when $p^{-1}(U)$ is open in $X$. It is the finest topology on the quotient making $p$ continuous.

Every map has $f(x)=f(x)$, so the relation is reflexive. Equality is symmetric, so $f(x)=f(y)$ for all the specified maps is equivalent to $f(y)=f(x)$; the relation is symmetric. If $x\mathcal{R}y$ and $y\mathcal{R}z$, every continuous map to every Hausdorff space has $f(x)=f(y)=f(z)$, proving $x\mathcal{R}z$. Thus $\mathcal{R}$ is an equivalence relation.

If $x\in A=g^{-1}(B)$ and $y\mathcal{R}x$, the definition of the relation applies in particular to the fixed continuous map $g$ into the fixed Hausdorff space $Y$. It gives $g(y)=g(x)\in B$, hence $y\in A$. Thus $A$ is a union of whole equivalence classes, also called a saturation of a subset.

Always $A\subseteq p^{-1}(p(A))$. Conversely, a point in $p^{-1}(p(A))$ shares its class with some $x\in A$, and the preceding argument places it in $A$. Hence

Take distinct classes $p(x)\ne p(y)$. The failure of $x\mathcal{R}y$ supplies a Hausdorff space $Y$ and a continuous map $g:X\to Y$ with $g(x)\ne g(y)$. Choose disjoint open neighborhoods $V,W$ of these two images. Their inverse images $A=g^{-1}(V)$ and $B=g^{-1}(W)$ are disjoint open subsets of $X$, and part (ii) says both are saturated.

The quotient topology therefore makes $p(A)$ and $p(B)$ open, since their inverse images are precisely $A,B$. They are disjoint as well: an equivalence class meeting both would, by saturation, be contained in both. They contain $p(x)$ and $p(y)$ respectively. Thus $X/\mathcal{R}$ is Hausdorff. This construction is the Hausdorff reflection by continuous maps.