Exercise 18, § 4

(a) Let Y be a completely regular space. Show that the following conditions are equivalent:

(α) Every prime ideal of $\mathscr{C}(Y; \mathbf{R})$ is maximal. (β) Every countable intersection of open subsets of Y is open. (γ) Every continuous real-valued function on Y is locally constant. (δ) The ring $\mathscr{C}(Y; \mathbf{R})$ is absolutely flat. (Use Exercise 15 (f) of § 2.)

If these hold, the space $X = \mathrm{Spec}(\mathscr{C}(Y; \mathbf{R}))$ is canonically identified with the Stone-Čech compactification $\beta Y$ of Y. Such a space Y is called flat.

(b) Every subspace of a flat space is flat. Every completely regular quotient space of a flat space is flat. Every finite product of flat spaces is flat. Every sum of flat spaces is a flat space.

(c) In a flat space, every countable subspace is closed and discrete. In particular every countable flat space is discrete.

(d) Every Weierstrassian completely regular space (General Topology, Chapter IX, § 1, Exercise 22) (and a fortiori every compact space), which is flat, is finite.

(e) The space associated with a non-trivial ultrafilter is extremely disconnected (General Topology, Chapter I, § 11, Exercise 21) but is not flat.

(f) The discrete space $\mathbf{N}$ is flat but its Stone-Čech compactification is not flat and the spectra of the rings $\mathscr{C}(\mathbf{N}; \mathbf{R})$ and $\mathscr{C}^\infty(\mathbf{N}; \mathbf{R})$ are therefore not isomorphic.