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. It seems like every space is a subspace of a product $\prod_{i \in I} \mathbb{S}$.

canembed all spaces in $\prod_{i \in I} \Bbb S'$ products. $\endgroup$