Construction of a Hausdorff space from a topological space

Let $HX$ be the quotient space $X/\sim$, where $x\sim y$ iff $f(x)=f(y)$ for each $f$ from $X$ to a Hausdorff space. Let $q:X\to HX$ be the quotient map.

Let us show that $HX$ is Hausdorff: Take $[x]\ne[y]\in HX$, i.e. for each $x,y$ representing these classes there is a map $f:X\to Y$ into a Hausdorff space $Y$ such that $f(x)\ne f(y)$. There are disjoint open sets $f(x)\in U,f(y)\in V$. Then $f^{-1}(U)$ and $f^{-1}(V)$ are disjoint open neighborhoods of $x$ and $y$. Assume that $z\in f^{-1}(U)$ and $z\sim v$. Then by definition of '$\sim$' we have $f(z)=f(v)$, so we conclude that $f^{-1}(U)$ and $f^{-1}(V)$ are $\sim$-saturated disjoint open sets. It follows that $q(f^{-1}(U))$ and $q(f^{-1}(V))$ are disjoint open neighborhoods of $[x]$ and $[y]$.

Now, assume that $f:X\to Y$ is a continuous map into a Hausdorff space. Whenever $x\sim y$ we also have $f(x)=f(y)$, hence there is a unique induced map $\tilde f:HX\to Y$ such that $\tilde f\circ q=f$.

This shows that the category $\mathbf{Top}_2$ of Hausdorff spaces is a full reflective subcategory of $\mathbf{Top}$.

What you are saying is that the full subcategory of Hausdorff spaces is a reflective subcategory of the category of topological spaces. That is to say, the inclusion functor $i: Haus \to Top$ has a left adjoint $H: Top \to Haus$, which is sometimes called Hausdorffification.

One way to see there exists such a thing is by the general adjoint functor theorem, whose hypothesis are easily checked in this case.

There are various constructions for the maximal Hausdorff quotient, see MO/78175 and MO/11191. I quite like the transfinite construction. Actually this is a special case of Kelly's paper on transfinite constructions, see also the corresponding nlab article.