The Stacks project

5.22 Profinite spaces

Here is the definition.

Definition 5.22.1. A topological space is profinite if it is homeomorphic to a limit of a diagram of finite discrete spaces.

This is not the most convenient characterization of a profinite space.

Lemma 5.22.2. Let $X$ be a topological space. The following are equivalent

  1. $X$ is a profinite space, and

  2. $X$ is Hausdorff, quasi-compact, and totally disconnected.

If this is true, then $X$ is a cofiltered limit of finite discrete spaces.

Proof. Assume (1). Choose a diagram $i \mapsto X_ i$ of finite discrete spaces such that $X = \mathop{\mathrm{lim}}\nolimits X_ i$. As each $X_ i$ is Hausdorff and quasi-compact we find that $X$ is quasi-compact by Lemma 5.14.5. If $x, x' \in X$ are distinct points, then $x$ and $x'$ map to distinct points in some $X_ i$. Hence $x$ and $x'$ have disjoint open neighbourhoods, i.e., $X$ is Hausdorff. In exactly the same way we see that $X$ is totally disconnected.

Assume (2). Let $\mathcal{I}$ be the set of finite disjoint union decompositions $X = \coprod _{i \in I} U_ i$ with $U_ i$ nonempty open (and closed) for all $i \in I$. For each $I \in \mathcal{I}$ there is a continuous map $X \to I$ sending a point of $U_ i$ to $i$. We define a partial ordering: $I \leq I'$ for $I, I' \in \mathcal{I}$ if and only if the covering corresponding to $I'$ refines the covering corresponding to $I$. In this case we obtain a canonical map $I' \to I$. In other words we obtain an inverse system of finite discrete spaces over $\mathcal{I}$. The maps $X \to I$ fit together and we obtain a continuous map

\[ X \longrightarrow \mathop{\mathrm{lim}}\nolimits _{I \in \mathcal{I}} I \]

We claim this map is a homeomorphism, which finishes the proof. (The final assertion follows too as the partially ordered set $\mathcal{I}$ is directed: given two disjoint union decompositions of $X$ we can find a third refining both.) Namely, the map is injective as $X$ is totally disconnected and hence $\{ x\} $ is the intersection of all open and closed subsets of $X$ containing $x$ (Lemma 5.12.11) and the map is surjective by Lemma 5.12.6. By Lemma 5.17.8 the map is a homeomorphism. $\square$

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. $\square$

Lemma 5.22.4. Let $X$ be a profinite space. Every open covering of $X$ has a refinement by a finite covering $X = \coprod U_ i$ with $U_ i$ open and closed.

Proof. Write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a limit of an inverse system of finite discrete spaces over a directed set $I$ (Lemma 5.22.2). Denote $f_ i : X \to X_ i$ the projection. For every point $x = (x_ i) \in X$ a fundamental system of open neighbourhoods is the collection $f_ i^{-1}(\{ x_ i\} )$. Thus, as $X$ is quasi-compact, we may assume we have an open covering

\[ X = f_{i_1}^{-1}(\{ x_{i_1}\} ) \cup \ldots \cup f_{i_ n}^{-1}(\{ x_{i_ n}\} ) \]

Choose $i \in I$ with $i \geq i_ j$ for $j = 1, \ldots , n$ (this is possible as $I$ is a directed set). Then we see that the covering

\[ X = \coprod \nolimits _{t \in X_ i} f_ i^{-1}(\{ t\} ) \]

refines the given covering and is of the desired form. $\square$

Lemma 5.22.5. Let $X$ be a topological space. If $X$ is quasi-compact and every connected component of $X$ is the intersection of the open and closed subsets containing it, then $\pi _0(X)$ is a profinite space.

Proof. We will use Lemma 5.22.2 to prove this. Since $\pi _0(X)$ is the image of a quasi-compact space it is quasi-compact (Lemma 5.12.7). It is totally disconnected by construction (Lemma 5.7.9). Let $C, D \subset X$ be distinct connected components of $X$. Write $C = \bigcap U_\alpha $ as the intersection of the open and closed subsets of $X$ containing $C$. Any finite intersection of $U_\alpha $'s is another. Since $\bigcap U_\alpha \cap D = \emptyset $ we conclude that $U_\alpha \cap D = \emptyset $ for some $\alpha $ (use Lemmas 5.7.3, 5.12.3 and 5.12.6) Since $U_\alpha $ is open and closed, it is the union of the connected components it contains, i.e., $U_\alpha $ is the inverse image of some open and closed subset $V_\alpha \subset \pi _0(X)$. This proves that the points corresponding to $C$ and $D$ are contained in disjoint open subsets, i.e., $\pi _0(X)$ is Hausdorff. $\square$


Comments (4)

Comment #5552 by Curious dilettante on

What is the condition of being cofiltered in Lemma 0ET8 required for? I attempted to carry out the proof and it appears that I did not use this condition. I mean, two different points in the limit are separated by a map to a space of the diagram and then you can lift the desired properties of separation ( and totally disconnected) from that space to the limit, right?

Comment #5736 by on

OK, yes, thanks. I added your suggested improvement with your proof. Hopefully correct. See this.

Comment #6209 by Amnon Yekutieli on

The spaces in condition (2) are called Stone spaces in topology literature.

I think (based on recent investigations, and on reading the Scholze-Clausen condensed math notes) that "limit" and "cofiltered limits" should be replaced by "inverse limit of finite discrete spaces". In other words, there is an inverse system (in the most naive sense of directed sets) of finite discrete spaces , , and .

Also, it seems that the set of partitions of a Stone space X into finite disjoint unions of closed (and open) sets is actually codirected (the opposite of directed set), by the relation of refinement. This codirected set accounts for all finite discrete quotients of X. Thus X is the inverse limit of this inverse system.

This is analogous to "open subgroups of finite index" in the case of profinite abelian groups.

Furtheremore, the caegory of Stone spaces (or profinite sets) seems to be equivalent to Pro(Set_{fin}), the cat of pro-objects of the cat of finite sets.

Amnon

Comment #6210 by on

@#6209 In the stacks project we use colimits and limits to distinguish between "projective limits" and "direct limits" and we do not use this terminology. See Section 4.14. Having said this, I think what you say in your second paragraph is the content ot Lemma 5.22.2. What you say in your third paragraph follows easily from Lemmas 5.22.4 and 5.22.2; we just have a different order of the arguments. We don't have the characterization of the category of profinite spaces you mention in your fifth paragraph. I may add this the next time I go through all the comments. Thanks!


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