Lemma 5.22.3. A limit of profinite spaces is profinite.

**Proof.**
Let $i \mapsto X_ i$ be a diagram of profinite spaces over the index category $\mathcal{I}$. Let us use the characterization of profinite spaces in Lemma 5.22.2. In particular each $X_ i$ is Hausdorff, quasi-compact, and totally disconnected. By Lemma 5.14.1 the limit $X = \mathop{\mathrm{lim}}\nolimits X_ i$ exists. By Lemma 5.14.5 the limit $X$ is quasi-compact. Let $x, x' \in X$ be distinct points. Then there exists an $i$ such that $x$ and $x'$ have distinct images $x_ i$ and $x'_ i$ in $X_ i$ under the projection $X \to X_ i$. Then $x_ i$ and $x'_ i$ have disjoint open neighbourhoods in $X_ i$. Taking the inverse images of these opens we conclude that $X$ is Hausdorff. Similarly, $x_ i$ and $x'_ i$ are in distinct connected components of $X_ i$ whence necessarily $x$ and $x'$ must be in distinct connected components of $X$. Hence $X$ is totally disconnected. This finishes the proof.
$\square$

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