$X/{\sim}$ is Hausdorff if and only if $\sim$ is closed in $X \times X$

Solution 1:

Since the map $\pi:X\to X/\sim$ is open, it's clear that the map $g:X^2\to (X/\sim)^2$ given by $g(x,y)=(\pi(x),\pi(y))$ is open. What we claim is that $g(X^2-\sim)=(X/\sim)^2-\Delta_{X/\sim}$. Indeed, if $x\nsim y$ then $\pi(x)\ne\pi(y)$ which tells us that $g\left(X^2-\sim\right)\subseteq (X/\sim)^2-\Delta_{X/\sim}$. That said, if $(\pi(x),\pi(y))\notin\Delta_{X/\sim}$ then $\pi(x)\ne \pi(y)$ so that $x\nsim y$ so that $(x,y)\in X^2-\sim$ and clearly $g(x,y)=(\pi(x),\pi(y))$. Thus, $g(X^2-\sim)=(X/\sim)^2-\Delta_{X/\sim}$ as claimed. But, since $X^2-\sim$ is open by assumption, and $g$ is an open map we have that $(X/\sim)^2-\Delta_{X/\sim}$ is open, and so $\Delta_{X/\sim}$ is closed. This gives us $T_2$ness.

Solution 2:

Let $R$ be the subset of $X \times X$ which gives the equivalence relation $\sim$, and let $f\colon X \to X/{\sim}$ be the quotient map. Let $x, y \in X$ be points not equivalent under the relation, i.e. $(x, y) \notin R$. Since $R$ is closed and $X \times X$ has the product topology, there exist open sets $U, V$ in $X$ such that $(x, y) \in U \times V$ and $U \times V$ does not meet $R$. Can you separate $f(x)$ and $f(y)$ using $U$ and $V$? Remember that $f$ is assumed to be an open map.

[This is a lot like the proof of the fact that Alex is using: that a space $X$ is Hausdorff if and only if the diagonal is closed in $X \times X$.]

Solution 3:

Let $\pi:X\to X/\!\!\!\sim\;$ denote the projection map associated with $\sim$. (That is, for any $x\in X$, $\pi(x)$ is the $\sim$-equivalence class that $x$ belongs to.) Let $\nsim\; \subseteq X \times X$ be shorthand for the complement of $\;\;\sim\;\;$ in $X \times X\;$, i.e. $\nsim\;\;=\;(X \times X\;) \;\;-\; \sim\;$.

Suppose that $\pi(x) \neq \pi(y)\;$. (Here I'm relying on the fact that, since $\pi$ is surjective, any element $\widetilde{z}\in X/\!\!\!\sim\;$ may be written in the form $\pi(z)$, for some $z \in X$.) We must show that there exist open sets $U_{\pi(x)}, U_{\pi(y)} \subseteq X/\!\!\!\sim\;$ such that ${\pi(x)} \in U_{\pi(x)}$, ${\pi(y)} \in U_{\pi(y)}$, and $U_{\pi(x)} \cap U_{\pi(y)} = \varnothing\;$.

By assumption, $\;\sim\; \subseteq X \times X$ is closed, so $\nsim\; \subseteq X \times X$ is open. Therefore there exist open neighborhoods $N_x$ and $N_y$ of $x$ and $y$, respectively, such that $(x,\;y)\in N_x \times N_y \subseteq$$\;\;\nsim\;$. (This is because the family of all pairwise products of open subsets of $X$ is a basis for the product topology on $X \times X$.)

For any $v, w \in X$,

$$ (v,\;w) \;\in \;\nsim \;\;\;\;\;\Leftrightarrow\;\;\;\;\; \pi(v) \neq \pi(w)\;\;. $$

Therefore,

$$ N_x \times N_y \subseteq \;\;\nsim\;\;\;\;\Leftrightarrow\;\;\;\; \forall (v, w) \in N_x \times N_y \;[\pi(v) \neq \pi(w)] \;\;\;\;\Leftrightarrow\;\;\;\; \pi[N_x] \cap \pi[N_y] = \varnothing $$

Furthermore, since $\pi$ is open (by assumption), the image sets $\pi[N_x], \pi[N_y] \subseteq X/\!\!\!\sim\;$ are open neighborhoods of ${\pi(x)}$ and ${\pi(y)}$, respectively. Therefore, $\pi[N_x]$ and $\pi[N_y]$ are the desired open neighborhoods $U_{\pi(x)} \ni {\pi(x)}, U_{\pi(y)} \ni {\pi(y)}$.

Solution 4:

Let $Y$ be the quotient. This is a question about the diagonal $D$ in $Y\times Y$.

Note that the product map $X\times X \to Y \times Y$ is also a quotient map (this is where open-ness is needed), and that your set $\sim$ is precisely $(\pi\times\pi)^{-1}(D)$. Thus $\sim$ is closed if and only if $D$ is closed.

Now, $D$ is closed if and only if $Y$ is Hausdorff, for any topological space $Y$. This follows by considering disjoint neighborhoods $U$ and $V$ of any two distinct points $p$ and $q$: $(p,q)\in U\times V$ which is open and disjoint from $D$, so $D^c$ is closed. Conversely, the closedness of $D$ implies the existence of some product open set $U\times V$ disjoint from $D$ and containing $(p,q)$, and thus $U$ and $V$ are open sets separating $p$ and $q$.