I think I understand the mechanism of constructing Stone-Čech well from Wikipedia.
However, I fail when trying to connect this with any concrete examples. For example, for the simplest examples, like $\mathbb{R}$, how do we construct $\beta \mathbb{R}$? I really couldn´t find any explicit construction anywhere, so any example would be appreciated.
I have read that "we do not construct Stone-Čech compactification. we just define it and prove it exists", but I still want to believe, haha.
Best Answer
Only for very rare spaces ($\omega_1$ in the order topology is one) do we have a concrete space that is provably the Cech-Stone compactification of $X$. We cannot answer many questions on $\beta \Bbb N$ (the naturals as a countable discrete space) in standard set theory. For that space, as well as $\beta \Bbb R$ there are many papers with some results (we know some things in some models of set theory), van Douwen, van Mill, KP Hart, Comfort, Kunen and many others have written on the C-S compactifications of these spaces and still much is unknown. We "know" (believe) it exists because the Axioms of Choice tells us this, but we only have a "fuzzy image" of it. It's a useful but elusive object.