The Stacks project

Example 61.6.3. Let $k$ be a field. Let $T$ be a profinite topological space. There exists an ind-Zariski ring map $k \to A$ such that $\mathop{\mathrm{Spec}}(A)$ is homeomorphic to $T$. Namely, just apply Lemma 61.6.2 to $T \to \pi _0(\mathop{\mathrm{Spec}}(k)) = \{ *\} $. In fact, in this case we have

\[ A = \mathop{\mathrm{colim}}\nolimits \text{Map}(T_ i, k) \]

whenever we write $T = \mathop{\mathrm{lim}}\nolimits T_ i$ as a filtered limit with each $T_ i$ finite.

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