Resource request: Stone-Čech construction generalized I was looking at the wikipedia article on the Stone-Čech compactification, and read that...

The Stone–Čech construction can be performed for more general spaces $X$, but the map $X \to \beta X$ need not be a homeomorphism to the image of $X$ (and sometimes is not even injective).

Despite $X \to \beta X$ possibly not being an embedding for general $X$, the article says functoriality and the same universal property will hold in the general case, and since I am currently self-studying categorical adjunctions, I think I would like to see the general "Stone-Čech construction".
After searching online, though, I can't seem to find a resource that explains a construction for non-Tychonoff spaces. Can anyone tell me where I can find a general construction - maybe in lecture notes, a free online resource, or a part of a book, etc?
 A: I think the following construction technique still works (which I think is the original Tychonoff one): for any space $X$ let $\mathcal{F}$ be 
the set $\mathcal{F} = \{f: X \to [0,1]: f \text{ continuous}\}$. This a well-defined set and non-empty  (which could consist of only constant the functions for some spaces $X$, but these are always in $\mathcal{F}$.)
Define $P_X = [0,1]^{\mathcal{F}}$ as the set $\{g: \mathcal{F} \to [0,1]\}$ in the product topology, so that it has the smallest topology that makes all $\pi_f: P_X \to [0,1]$ continuous (where $\pi_f(g) = g(f)$). By Tychonoff's theorem, $P_X$ is a compact space (it's just a product of many copies of $[0,1]$ which is compact) and it's also Hausdorff (all product of Hausdorff spaces are).
Then define $e_X: X \to P_x$ (the evaluation map, a.k.a Gelfand transform, I believe) by $(e_X(x))(f) = f(x)$ for all $f \in \mathcal{F}$, note that $e_X(x)$ should be a member of the product, so a function $\mathcal{F} \to [0,1]$, so we specify what its value is on each $f$.  
By the universal property of maps into the product, $e_X$ is a continuous function of $X$ to $P_X$, as $\pi_f(e) = f$ for all $f \in \mathcal{F}$. For Tychonoff spaces $e_X$ is an embedding by the theorem: $e_X$ is an embedding into $P_X$ iff 


*

*$\mathcal{F}$ separates points: $\forall x \neq y \in X: \exists f \in \mathcal{F}: f(x) \neq f(y)$.

*$\mathcal{F}$ separates points from closed sets: $\forall x \in X: \forall C \subseteq X: x \notin C \text{ and } C \text{ closed} \implies \exists f \in \mathcal{F}: f(x) \notin \overline{f[C]}$.
The first condition is just a restatement of injectivity. The second ensures that the inverse (which then exists) from $e[X]$ to $X$ is continuous. This argument works for all spaces. And $\mathcal{F}$ obeys these conditions for a space $X$, iff $X$ is Tychonoff. (We can state and prove this embedding theorem for any family of maps, and the product of the codomains, mutatis mutandis.)
$\beta X:= \overline{e_X[X]} \subseteq P_X$ and $e = e_X: X \to P_X$ considered as a map to $\beta X$ (codomain restriction, which does not impact continuity).
Then $\beta X$ is compact Hausdorff, as a closed subset of the compact $P$, so $\beta X$ lies in the right category, and we need to see that the pair $(X,e)$ is universal for maps to compact Hausdorff spaces.
The proof of the universal property (existence part) uses that all compact Hausdorff $K$ embed into a product $P_K=[0,1]^{\mathcal{F}'}$ of copies of $[0,1]$ as well, using the above construction with $\mathcal{F}'$ as the set of continuous maps from $K$ into $[0,1]$, as all such $K$ are Tychonoff (even $T_4$). So the map $e_K: K \to P_K$ for some product of $[0,1]$ is a homeomorphism when restricted as $e_K: K \to e_K[K] = \beta K$ (the latter as $e_K[K]$ is already compact in the Hausdorff product $P_K$, so closed, so its closure, i.e. $\beta K$, is just $e_K[K]$ itself.
Now, if $f: X \to K$ is any continuous map from $X$ to a compact Hausdorff $K$: consider the following: For every continuous map $f': K \to [0,1]$, we have that $f' \circ f: X \to [0,1]$, is also continuous, so lies in $\mathcal{F}$. Define $h$ from $P_X$ to $P_K$ by $h(g)(f') = g(f' \circ f)$, where $g \in P_X = [0,1]^{\mathcal{F}}$, and $f' \in \mathcal{F}'$. Then check that $h' = h|_{\beta X} \circ e_X$ maps $X$ into $e_K[K]$, as $h'(x) = e_K[f(x)]$ for all $x \in X$. Then define $\beta f: X \to K: \beta f := e_K^{-1} \circ h'$ which is continuous as $e_K$ is a homeomorphism, and note that $\beta f \circ e = f$, as required. The unicity is similar, as the condition $\beta f \circ e = f$ forces what $\beta f$ has to be on $e[X]$, and this is thus uniquely determined on the closure (as $K$ is Hausdorff and $e[X]$ is by definition dense in $\beta X$).
There are probably books that do it this way too, but I wouldn't be able to name one. All that I checked (Engelking, Herrlich, and some more) only worked within Tychonoff spaces, and then $(\beta X, e)$ is a compactification. For me, this is more natural, but the construction still works in general. Note that if $X$ only has constant continuous maps to $[0,1]$, $\beta X$ is just a singleton, and $e$ is the unique constant map to it. So it's not as intuitive any more.
I found one online reference,  in Tom Leinster's book "basic category theory" where he discusses a general theorem (Special Adjoint Functor Theorem, SAFT). In 6.3.14 he derives the existence of a left adjoint for the forgetful functor from $\textrm{CompHaus}$ to $\textrm{Top}$, which is the Stone-Cech construction. He does mention there the construction as the closure of the image of $X$ under the canonical map from $X$ into $[0,1]^{C(X,[0,1])}$, but does not mention whether this always holds or only for Tychonoff spaces (which he calls "a mild separation property").. 
Finally, this nice blog post explicitly says the construction I just gave is the general one that works for all $X$... I don't know whether this counts as a "reference", though (it has the same status as this answer in a way).
Intuitively, we'd like use a "set" $\mathbb{C} = \{f: X \to K\}$ where $f$ is continuous and $K$ ranges over all compact Hausdorff (cH) spaces, and define an $e_X$ as before into a product over all cH spaces, and then $f$ factors over $e_X$ by $\pi_f$ essentially. This doesn't really work, as we get proper classes, but luckily $[0,1]$ is a so-called "generator'" for this class cH, so maps into $[0,1]$ suffice.
