Let $\mathbb{S}$ be the Sierpiński space, the two pointed space $\{ 0, 1 \}$ with open sets $\{0 \}$, $\emptyset$, $\{ 0, 1 \}$. We give $\{ 0, 1 \}$ a partial order where $0 < 1$.


Let $X$ be a topological space. Consider the space $Y = \prod\_{f : X \rightarrow \mathbb{S}} \mathbb{S}$. This space is compact by Tychonoff's theorem.


There is a natural map $f : X \rightarrow Y$ and the closure of the image of $f$ in $Y$, $X\_0$. This is a closed subspace of a compact space and so it is compact.


Is $X\_0$ the Stone–Čech compactification of $X$ under certain conditions? I suspect that it always is. However, some have told me that this construction only works for $T\_{ 3 \frac{1}{2}}$ spaces. I would find it quite helpful if someone could link me to a source explaining the limitations of this construction if there are some. Alternatively, I think there is an adjoint functor theorem in terms of cogenerators? See the special adjoint functor theorem at [nLab](http://ncatlab.org/nlab/show/adjoint+functor+theorem#statement). It seems like every space is a subspace of a product $\prod\_{i \in I} \mathbb{S}$.



Is this space the Stone–Čech compactification?
No, the closure of the image of $f$ in $Y$ is never the Stone-Čech compactification of $X$ unless $X$ is empty. In particular, consider the element $a\in Y$ which is $1$ on every coordinate. Note that the only open subset of $Y$ that contains $a$ is $Y$ itself. So, if $X$ is nonempty, then $a$ will be in the closure $X\_0$ of the image of $f$. This means that $X\_0$ contains a point whose only neighborhood is the whole space $X\_0$. Also, $a$ is not in the image of $f$ (no element of $X$ is in every closed subset of $X$), so $a$ is not the only point of $X\_0$. This means $X\_0$ is not $T\_1$, so it cannot be the Stone-Čech compactification of $X$.


The main moral here is that the Stone-Čech compactification is not about making a space compact: it is about making a space compact *Hausdorff*. So if you have some sort of universal construction that does not enforce the Hausdorffness condition, there is no reason to expect to get the Stone-Čech compactification.


