Stone–Čech compactification and AC

Question

So I just started reading on Tychonoff spaces and Stone–Čech compactification. Could someone recommend me a good reference with the proof that the axiom of choice implies that every topological space has a Stone–Čech compactification?

Answer 1

In Howard and Rubin ("Consequences of the Axiom of Choice"), they give form [14L]:

For every topological space $X$ there is a pair $(Y,j)$ such that $Y$ is a compact topological space and $j$ is a continuous surjection from $X$ onto $Y$ such that for all topological spaces $Z$ such that $Z$ is compact and completely regular and for all continuous $f :X \to Z$, there is a unique continuous $h: Y \to Z$ such that $f = h \circ j$.

They give the reference Rubin H./Scott. D "Some topological theorems equivalent to the Boolean prime ideal theorem", Bull. Amer. Math. Soc. 60, 389 for this. So this fact is equivalent to the Boolean Prime Ideal Theorem (which also is equivalent to the theorem that a product of compact Hausdorff spaces is compact, while the full axiom of choice is equivalent to the theorem that a product of compact spaces is compact.) So it is implied by AC but a bit weaker than it.

Personally, I'm not sure that $j$ being a surjection is actually true (for Tychonoff spaces it's even an embedding), but I just copied their statement. If I follow my own proof of existence from this answer for general spaces, I get that $Y$ is compact Hausdorff and $j$ just continuous but not onto. I do use that products of $[0,1]$ are compact (this is where AC is used). Maybe someone has an idea of how to see their version and its relation to my (standard I think) construction? I cannot consult the original paper to check.