A submanifold $\{(p,q) \in M \times M: p \sim q\}$ is closed if and only if $M/\sim$ has the Hausdorff property?

Question

For a manifold $M$, if $M/\sim$ has the structure of a possibly non-Hausdorff manifold, such that the quotient map $\pi: M \to M/\sim$ is a submersion. Then I want to show that the set $R = \{(p,q) \in M \times M: p \sim q\}$ is a submanifold of $M \times M$, and it is closed if and only if $M/\sim$ has the Hausdorff property. Is anyone able to show me the proof?

Answer 1

You can replace manifold with locally compact space I think.

Facts

• If $M$ is locally compact and $\pi: M \to M / \sim$ is a quotient map then $\pi \times \pi : M \times M \to M \times M / (\sim \times \sim)$ is a quotient map.

• A space $M$ is Hausdorff if and only if its diagonal $\Delta_M \subseteq M \times M$ is closed.

Then if $\sim \, \subseteq M \times M$ denotes the set of points $(x,y)$ with $x \sim y$, we have $$ \sim \,= (p \times p)^{-1}(\Delta_{M/ \sim}) $$ By definition of quotient topology and the previous facts we have that $\sim$ is closed iff $M/ \sim$ is Hausdorff.

On the simpler side of things you can take $x,y$ such that $x \not\sim y$ and take an open $U$ of the form $U_1 \times U_2$ set containing $(x,y)$ such that $U \cap \sim = \varnothing$ and take the image of those sets.