Construction of a Hausdorff space from a topological space Let $X$ be a topological space. Is there a Hausdorff space $HX$ and a continuous function $i:X\rightarrow HX$ such that for any Hausdorff space $A$ and a continuous function $j:X\rightarrow A$, there exists a unique continuous function $f:HX\rightarrow A$ satisfying  $fi=j$.

$$\begin{array}{ccccccccc} X & \xrightarrow{i} & HX & \\\ &
 \searrow{j} & \downarrow{f} \\&&A \end{array}$$

Note:  I heard before that free objects and "left adjoints" are really the same thing (not sure). Since I don't know the definition of left adjoints and did not study category theory yet, I chose to phrase my question in the way I am most comfortable with.
Thank you
 A: 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.
A: 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 $\mathbf{Top}$.
A: 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.
A: A construction suggested on Wikipedia in fact produces a left adjoint for the forgetful functor $\mathcal{H}\to Top$ for any full subcategory $\mathcal{H}$ of $Top$ consisting of Hausdorff spaces that is closed under taking products and taking closed subspaces - for example $\mathcal{H}=Haus$ itself, or $\mathcal{H}=\mathcal{T_{3\frac{1}{2}}}$ (completely regular Hausdorff spaces), or $\mathcal{H}=CompHaus$ (compact Hausdorff spaces, the case to which Wikipedia applies the construction to find the Stone-Čech compactification).
Indeed, if $f:X\to Y$ is a continuous map with $Y$ Hausdorff, and $f(X)$ is dense in $Y$, there is an injection $y\mapsto\{f^{-1}(U)\mid y\in U \,\&\,U \text{ open in }Y\}$ from $Y$ into the double power set $\mathcal{P}^{2}(X)=\mathcal{P}(\mathcal{P}(X))$ of $X$.
So for $X\in Top$, let $\mathcal{F}(X)$ denote the set of triples $(Y,\tau,f)$ with $Y\in\mathcal{P}^{3}(X)$, $\tau$ is a topology on $Y$ such that $(Y,\tau)\in\mathcal{H}$, and $f:X\to (Y,\tau)$ is continuous with dense image. And let $g:X\to\prod_{(Y,\tau,f)\in\mathcal{F}(X)}(Y,\tau)$ be the natural map. Then the closure $\mathcal{G}(X):=\overline{g(X)}$ of the image of $g$ is in $\mathcal{H}$, and the corestriction $g:X\to\mathcal{G}(X)$ of $g$ to $\mathcal{G}(X)$ is universal among continuous maps from $X$ to objects of $\mathcal{H}$.
In the case $\mathcal{H}=Haus$, by universality $g:X\to\mathcal{G}(X)$ is just $nat:X\to HX$, which is surjective. It follows that here the image of $g$ is closed in $\prod_{(Y,\tau,f)\in\mathcal{F}(X)}(Y,\tau)$, a fact not immediately clear from the definitions.
