Like Alison said, one can identify $Spec(k^{\mathbb{N}})$ with the set of ultrafilters on $\mathbb{N}$. There is a canonical topology on this set, which makes it into the Stone-Cech compactification of $\mathbb{N}$, $S \mathbb{N}$: one takes as a basis the sets $U_A = {${$F \in S \mathbb{N} : A \in F}$F $} , where$A \subset $\mathbb{N}$. \mathbb{N}$.$S \mathbb{N}$is universal among compactifications of$ \mathbb{N}$, in the sense that every map from$\mathbb{N}$to compact$X$extends uniquely to$S\mathbb{N}$. It's not hard to see that$S \mathbb{N}$is homeomorphic to$(Spec(k^{\mathbb{N}}), zariski)$. I think that what Mumford is pointing at are Stone-Cech compactifications in general rather than$Spec(k^{\mathbb{N}})$in particular. There's quite a good brief definition and explanation of them in Steen & Seebach's Counterexamples in Topology'; you could also check out the references on the Wikipedia page. 1 Like Alison said, one can identify$Spec(k^{\mathbb{N}})$with the set of ultrafilters on$\mathbb{N}$. There is a canonical topology on this set, which makes it into the Stone-Cech compactification of$\mathbb{N}$,$S \mathbb{N}$: one takes as a basis the sets$U_A = {F \in S \mathbb{N} : A \in F}$, where A \subset$\mathbb{N}$.$S \mathbb{N}$is universal among compactifications of$ \mathbb{N}$, in the sense that every map from$\mathbb{N}$to compact$X$extends uniquely to$S\mathbb{N}$. It's not hard to see that$S \mathbb{N}$is homeomorphic to$(Spec(k^{\mathbb{N}}), zariski)$. I think that what Mumford is pointing at are Stone-Cech compactifications in general rather than$Spec(k^{\mathbb{N}})\$ in particular. There's quite a good brief definition and explanation of them in Steen & Seebach's Counterexamples in Topology'; you could also check out the references on the Wikipedia page.