Fill gaps in a topology proof

Question

I am trying to fill gaps in a proof located at this page by Todd Trimble.

I prefer the proofs by citing "Stone Spaces" book (and maybe Wikipedia).

Let $X_1,\dots,X_n$ be sets.

• Todd claims that topologies on $\beta X_1 \times \ldots \times \beta X_n$ are precisely compact Hausdorff spaces on $\beta X_1 \times \ldots \times \beta X_n$. How does this follow? I suppose we need first prove that topologies on $\beta X_i$ are precisely compact Hausdorff spaces on $\beta X_i$, and second prove that finite products of compact Hausdorff spaces are precisely compact Hausdorff spaces. Right? Also: https://en.wikipedia.org/wiki/Stone_space seems to claim that boolean algebras are equivalent to compact totally disconnected Hausdorff spaces (not to compact Hausdorff spaces), so do I misunderstant something?

Answer 1

A "relation from $\beta X$ to $\beta Y$ in the pretopos of compact Hausdorff spaces" is just a subobject of the product $\beta X\times\beta Y$ in the category of compact Hausdorff space. Concretely, that's just a subspace of $\beta X\times \beta Y$ which is itself a compact Hausdorff space, or in other words just a closed subspace of $\beta X\times\beta Y$.

There is no assertion being made here about general topologies or compact Hausdorff spaces. All that is being said is that there is a natural bijection between topogenies from $X$ to $Y$ and closed subspaces of $\beta X\times\beta Y$.