# Quotient space of a Hausdorff space is also a Hausdorff space

I am trying to learn topology but I dont know how to proof this problem.

Let $$X$$ be a Hausdorff space, ~ an equivalence relation and $$\pi:X \to X/{\sim}$$ the canonical map. $$X/{\sim}$$ is also Hausdorff, if there exists a continuous function $$s:X/{\sim} \to X$$, such that $$\pi \circ s=\textrm{Id}_{X/{\sim}}$$

I've read that the diagonal of a Hausdorff space is closed regarding the product topology but I don't know how to proceed or where to start

I would go about it this way (letting $$Y$$ represent the quotient space $$X/\sim$$):
• First, use the fact that $$\pi\circ s$$ is the identity on $$Y$$ (so is one-to-one) to prove that $$s$$ is one-to-one.
• Next, letting $$Z$$ be the image of $$s,$$ note that $$Z$$ is a subspace of $$X,$$ so is Hausdorff.
• Finally, show that $$\pi\restriction Z$$ is the inverse of $$s.$$ Since both $$s$$ and $$\pi\restriction Z$$ are continuous, that means that $$s$$ is a homeomorphism from $$Y$$ to $$Z,$$ so since $$Z$$ is Hausdorff, so is $$Y.$$
Notice that $$s:X/_\sim\to \operatorname{im}s$$ and $$\left.\pi\right\rvert_{\operatorname{im} s}:\operatorname{im} s\to X/_\sim$$ are continuous maps, and they are one the inverse of the other (this is true set-theoretically speaking). Therefore, $$X/_\sim$$ is homeomorphic to $$\operatorname{im}s\subseteq X$$, which inherits the Hausdorff property.