The following two theorems seem to be well-known:
Theorem 1: Suppose $X$ is a compact and Hausdorff space and $\sim$ is an equivalence relation on $X$. Let $R=\{(x_1,x_2)\in X\times X|x_1\sim x_2\}$. Suppose $R$ is a closed subset of $X\times X$, then the quotient space $X/\!\sim$ is Hausdorff.
Theorem 2: Suppose $X$ is a compact metric space and $\sim$ is an equivalence relation on $X$. If the quotient space $X/\!\sim$ is Hausdorff, then $X/\!\sim$ is metrizable.
Can I ask if there are any references for these two results?
To be clear, I am not asking for the proofs of these statements. I am aware that both results have been discussed on StackExchange. A proof of Theorem 1 can be found here: Question about quotient of a compact Hausdorff space, and a proof of Theorem 2 can be found here: Quotient of compact metric space is metrizable (when Hausdorff)?. But it would be greatly appreciated if I may ask whether there are any references for these results in published works that can be directly cited. Thanks!
