Well, I was reading this article by Kelley and when reached the point where he say that $X_a$ is closed in $Y_a$ I had to stop, probably mine is just a stupid misunderstand but can't figure out how to prove that $X_a$ is closed.
Another problem: if $X_a$ is closed than $Y_a-X_a$ has to be open, thuse $Y_a-(Y_a-X_a)$ finite, and we picked some random space $X_a$ so that is not generally true.
This seems to be a typo (or a small mistake, if you prefer), and the proof goes along just fine if you also require that $\{\Lambda\}$ is open in each $X_a$. The space is still compact.
You can find the same proof, minus the typo, on the Wikipedia page of the Tychonoff's theorem.